Handbook of Differential Equations

Handbook of Differential Equations 3rd edition Daniel Zwillinger Academic Press, 1997

Contents Preface Introduction Introduction to the Electronic Version How to Use This Book I.A Definitions and Concepts 1 Definition of Terms . . . . . . . . . . . . . . . . 2 Alternative Theorems . . . . . . . . . . . . . . 3 Bifurcation Theory . . . . . . . . . . . . . . . . 4 A Caveat for Partial Differential Equations . . 5 Chaos in Dynamical Systems . . . . . . . . . . 6 Classification of Partial Differential Equations . 7 Compatible Systems . . . . . . . . . . . . . . . 8 Conservation Laws . . . . . . . . . . . . . . . . 9 Differential Resultants . . . . . . . . . . . . . . 10 Existence and Uniqueness Theorems . . . . . . 11 Fixed Point Existence Theorems . . . . . . . . 12 Hamilton-Jacobi Theory . . . . . . . . . . . . . 13 Integrability of Systems . . . . . . . . . . . . . 14 Internet Resources . . . . . . . . . . . . . . . . 15 Inverse Problems . . . . . . . . . . . . . . . . . 16 Limit Cycles . . . . . . . . . . . . . . . . . . . 17 Natural Boundary Conditions for a PDE . . . . 18 Normal Forms: Near-Identity Transformations 19 Random Differential Equations . . . . . . . . . 20 Self-Adjoint Eigenfunction Problems . . . . . . 21 Stability Theorems . . . . . . . . . . . . . . . . 22 Sturm-Liouville Theory . . . . . . . . . . . . . 23 Variational Equations . . . . . . . . . . . . . . 24 Well Posed Differential Equations . . . . . . . . 25 Wronskians and Fundamental Solutions . . . .

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2 15 19 27 29 36 43 47 50 53 58 61 65 71 75 78 83 86 91 95 101 103 109 115 119

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Zeros of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

I.B Transformations 27 Canonical Forms . . . . . . . . . . . . . . . . . . . 28 Canonical Transformations . . . . . . . . . . . . . 29 Darboux Transformation . . . . . . . . . . . . . . . 30 An Involutory Transformation . . . . . . . . . . . . 31 Liouville Transformation - 1 . . . . . . . . . . . . . 32 Liouville Transformation - 2 . . . . . . . . . . . . . 33 Reduction of Linear ODEs to a First Order System 34 Prufer Transformation . . . . . . . . . . . . . . . . 35 Modified Prufer Transformation . . . . . . . . . . . 36 Transformations of Second Order Linear ODEs - 1 37 Transformations of Second Order Linear ODEs - 2 38 Transformation of an ODE to an Integral Equation 39 Miscellaneous ODE Transformations . . . . . . . . 40 Reduction of PDEs to a First Order System . . . . 41 Transforming Partial Differential Equations . . . . 42 Transformations of Partial Differential Equations .

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128 132 135 139 141 144 146 148 150 152 157 159 162 166 168 173

II Exact Analytical Methods 43 Introduction to Exact Analytical Methods . . . . . . . . . . . . . 178 44 Look-Up Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 179 45 Look-Up ODE Forms . . . . . . . . . . . . . . . . . . . . . . . . . 219 II.A Exact Methods for ODEs 46 An Nth Order Equation . . . . . . . . . . . . . . . . . . 47 Use of the Adjoint Equation . . . . . . . . . . . . . . . . 48 Autonomous Equations - Independent Variable Missing 49 Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . 50 Clairaut’s Equation . . . . . . . . . . . . . . . . . . . . . 51 Computer-Aided Solution . . . . . . . . . . . . . . . . . 52 Constant Coefficient Linear Equations . . . . . . . . . . 53 Contact Transformation . . . . . . . . . . . . . . . . . . 54 Delay Equations . . . . . . . . . . . . . . . . . . . . . . 55 Dependent Variable Missing . . . . . . . . . . . . . . . . 56 Differentiation Method . . . . . . . . . . . . . . . . . . . 57 Differential Equations with Discontinuities . . . . . . . . 58 Eigenfunction Expansions . . . . . . . . . . . . . . . . . 59 Equidimensional-in-x Equations . . . . . . . . . . . . . . 60 Equidimensional-in-y Equations . . . . . . . . . . . . . . 61 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . 62 Exact First Order Equations . . . . . . . . . . . . . . . 63 Exact Second Order Equations . . . . . . . . . . . . . . 64 Exact Nth Order Equations . . . . . . . . . . . . . . . . 65 Factoring Equations . . . . . . . . . . . . . . . . . . . .

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224 226 230 235 237 240 247 249 253 260 262 264 268 275 278 281 284 287 290 292

66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Factoring Operators . . . . . . . . . . . . . . . . . . Factorization Method . . . . . . . . . . . . . . . . . Fokker-Planck Equation . . . . . . . . . . . . . . . . Fractional Differential Equations . . . . . . . . . . . Free Boundary Problems . . . . . . . . . . . . . . . . Generating Functions . . . . . . . . . . . . . . . . . . Green’s Functions . . . . . . . . . . . . . . . . . . . Homogeneous Equations . . . . . . . . . . . . . . . . Method of Images . . . . . . . . . . . . . . . . . . . Integrable Combinations . . . . . . . . . . . . . . . . Integral Representation: Laplace’s Method . . . . . . Integral Transforms: Finite Intervals . . . . . . . . . Integral Transforms: Infinite Intervals . . . . . . . . Integrating Factors . . . . . . . . . . . . . . . . . . . Interchanging Dependent and Independent Variables Lagrange’s Equation . . . . . . . . . . . . . . . . . . Lie Groups: ODEs . . . . . . . . . . . . . . . . . . . Operational Calculus . . . . . . . . . . . . . . . . . . Pfaffian Differential Equations . . . . . . . . . . . . . Reduction of Order . . . . . . . . . . . . . . . . . . . Riccati Equations . . . . . . . . . . . . . . . . . . . . Matrix Riccati Equations . . . . . . . . . . . . . . . Scale Invariant Equations . . . . . . . . . . . . . . . Separable Equations . . . . . . . . . . . . . . . . . . Series Solution . . . . . . . . . . . . . . . . . . . . . Equations Solvable for x . . . . . . . . . . . . . . . . Equations Solvable for y . . . . . . . . . . . . . . . . Superposition . . . . . . . . . . . . . . . . . . . . . . Method of Undetermined Coefficients . . . . . . . . . Variation of Parameters . . . . . . . . . . . . . . . . Vector Ordinary Differential Equations . . . . . . . .

II.B Exact Methods for PDEs 97 Backlund Transformations . . . . . 98 Method of Characteristics . . . . . 99 Characteristic Strip Equations . . 100 Conformal Mappings . . . . . . . . 101 Method of Descent . . . . . . . . . 102 Diagonalization of a Linear System 103 Duhamel’s Principle . . . . . . . . 104 Exact Equations . . . . . . . . . . 105 Hodograph Transformation . . . . 106 Inverse Scattering . . . . . . . . . . 107 Jacobi’s Method . . . . . . . . . . 108 Legendre Transformation . . . . . 109 Lie Groups: PDEs . . . . . . . . .

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294 300 303 308 311 315 318 327 330 334 336 342 347 356 360 363 366 379 384 389 392 395 398 401 403 409 411 413 415 418 421

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428 432 438 441 446 449 451 454 456 460 464 467 471

110 111 112 113 114 115 116

Poisson Formula . . . . . . . . . . . . Riemann’s Method . . . . . . . . . . . Separation of Variables . . . . . . . . . Separable Equations: Stackel Matrix . Similarity Methods . . . . . . . . . . . Exact Solutions to the Wave Equation Wiener-Hopf Technique . . . . . . . .

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478 481 487 494 497 501 505

III Approximate Analytical Methods 117 Introduction to Approximate Analysis . . . . . 118 Chaplygin’s Method . . . . . . . . . . . . . . . 119 Collocation . . . . . . . . . . . . . . . . . . . . 120 Dominant Balance . . . . . . . . . . . . . . . . 121 Equation Splitting . . . . . . . . . . . . . . . . 122 Floquet Theory . . . . . . . . . . . . . . . . . . 123 Graphical Analysis: The Phase Plane . . . . . 124 Graphical Analysis: The Tangent Field . . . . . 125 Harmonic Balance . . . . . . . . . . . . . . . . 126 Homogenization . . . . . . . . . . . . . . . . . . 127 Integral Methods . . . . . . . . . . . . . . . . . 128 Interval Analysis . . . . . . . . . . . . . . . . . 129 Least Squares Method . . . . . . . . . . . . . . 130 Lyapunov Functions . . . . . . . . . . . . . . . 131 Equivalent Linearization and Nonlinearization . 132 Maximum Principles . . . . . . . . . . . . . . . 133 McGarvey Iteration Technique . . . . . . . . . 134 Moment Equations: Closure . . . . . . . . . . . 135 Moment Equations: Ito Calculus . . . . . . . . 136 Monge’s Method . . . . . . . . . . . . . . . . . 137 Newton’s Method . . . . . . . . . . . . . . . . . 138 Pade Approximants . . . . . . . . . . . . . . . 139 Perturbation Method: Method of Averaging . . 140 Perturbation Method: Boundary Layer Method 141 Perturbation Method: Functional Iteration . . 142 Perturbation Method: Multiple Scales . . . . . 143 Perturbation Method: Regular Perturbation . . 144 Perturbation Method: Strained Coordinates . . 145 Picard Iteration . . . . . . . . . . . . . . . . . . 146 Reversion Method . . . . . . . . . . . . . . . . 147 Singular Solutions . . . . . . . . . . . . . . . . 148 Soliton-Type Solutions . . . . . . . . . . . . . . 149 Stochastic Limit Theorems . . . . . . . . . . . 150 Taylor Series Solutions . . . . . . . . . . . . . . 151 Variational Method: Eigenvalue Approximation 152 Variational Method: Rayleigh-Ritz . . . . . . . 153 WKB Method . . . . . . . . . . . . . . . . . . .

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510 511 514 517 520 523 526 532 535 538 542 545 549 551 555 560 566 568 572 575 578 582 586 590 598 605 610 614 618 621 623 626 629 632 635 638 642

IV.A Numerical Methods: Concepts 154 Introduction to Numerical Methods . . . . . 155 Definition of Terms for Numerical Methods 156 Available Software . . . . . . . . . . . . . . 157 Finite Difference Formulas . . . . . . . . . . 158 Finite Difference Methodology . . . . . . . . 159 Grid Generation . . . . . . . . . . . . . . . 160 Richardson Extrapolation . . . . . . . . . . 161 Stability: ODE Approximations . . . . . . . 162 Stability: Courant Criterion . . . . . . . . . 163 Stability: Von Neumann Test . . . . . . . . 164 Testing Differential Equation Routines . . .

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648 651 654 661 670 675 679 683 688 692 694

IV.B Numerical Methods for ODEs 165 Analytic Continuation . . . . . . . . . . . . . 166 Boundary Value Problems: Box Method . . . 167 Boundary Value Problems: Shooting Method 168 Continuation Method . . . . . . . . . . . . . 169 Continued Fractions . . . . . . . . . . . . . . 170 Cosine Method . . . . . . . . . . . . . . . . . 171 Differential Algebraic Equations . . . . . . . . 172 Eigenvalue/Eigenfunction Problems . . . . . . 173 Euler’s Forward Method . . . . . . . . . . . . 174 Finite Element Method . . . . . . . . . . . . 175 Hybrid Computer Methods . . . . . . . . . . 176 Invariant Imbedding . . . . . . . . . . . . . . 177 Multigrid Methods . . . . . . . . . . . . . . . 178 Parallel Computer Methods . . . . . . . . . . 179 Predictor-Corrector Methods . . . . . . . . . 180 Runge-Kutta Methods . . . . . . . . . . . . . 181 Stiff Equations . . . . . . . . . . . . . . . . . 182 Integrating Stochastic Equations . . . . . . . 183 Symplectic Integration . . . . . . . . . . . . . 184 Use of Wavelets . . . . . . . . . . . . . . . . . 185 Weighted Residual Methods . . . . . . . . . .

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698 701 706 710 713 716 720 726 730 734 744 747 752 755 759 763 770 775 780 784 786

IV.C Numerical Methods for PDEs 186 Boundary Element Method . . . . . . . . . . . . 187 Differential Quadrature . . . . . . . . . . . . . . 188 Domain Decomposition . . . . . . . . . . . . . . 189 Elliptic Equations: Finite Differences . . . . . . . 190 Elliptic Equations: Monte-Carlo Method . . . . . 191 Elliptic Equations: Relaxation . . . . . . . . . . 192 Hyperbolic Equations: Method of Characteristics 193 Hyperbolic Equations: Finite Differences . . . . . 194 Lattice Gas Dynamics . . . . . . . . . . . . . . .

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792 796 800 805 810 814 818 824 828

195 196 197 198 199

Method of Lines . . . . . . . . . . . . . . . Parabolic Equations: Explicit Method . . . Parabolic Equations: Implicit Method . . . Parabolic Equations: Monte-Carlo Method Pseudospectral Method . . . . . . . . . . .

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831 835 839 844 851

Preface When I was a graduate student in applied mathematics at the California Institute of Technology, we solved many differential equations (both ordinary differential equations and partial differential equations). Given a differential equation to solve, I would think of all the techniques I knew that might solve that equation. Eventually, the number of techniques I knew became so large that I began to forget some. Then, I would have to consult books on differential equations to familiarize myself with a technique that I remembered only vaguely. This was a slow process and often unrewarding; I might spend twenty minutes reading about a technique only to realize that it did not apply to the equation I was trying to solve. Eventually, I created a list of the different techniques that I knew. Each technique had a brief description of how the method was used and to what types of equations it applied. As I learned more techniques, they were added to the list. This book is a direct result of that list. At Caltech we were taught the usefulness of approximate analytic solutions and the necessity of being able to solve differential equations numerically when exact or approximate solution techniques could not be found. Hence, approximate analytical solution techniques and numerical solution techniques were also added to the list. Given a differential equation to analyze, most people spend only a small amount of time using analytical tools and then use a computer to see what the solution “looks like.” Because this procedure is so prevalent, this edition includes an expanded section on numerical methods. New sections on sympletic integration (see page 780) and the use of wavelets (see page 784) also have been added. In writing this book, I have assumed that the reader is familiar with differential equations and their solutions. The object of this book is not to teach novel techniques but to provide a handy reference to many popular techniques. All of the techniques included are elementary in the usual mathematical sense; because this book is designed to be functional it does not include many abstract methods of limited applicability. This handbook has been designed to serve as both a reference book and as a complement to a text on differential equations. Each technique described is accompanied by several references; these allow each topic to be studied in more detail. It is hoped that this book will be used by students taking courses in differential equations (at either the undergraduate or the graduate level). It will introduce the student to more techniques than they usually see in a differential equations

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class and will illustrate many different types of techniques. Furthermore, it should act as a concise reference for the techniques that a student has learned. This book should also be useful for the practicing engineer or scientist who solves differential equations on an occasional basis. A feature of this book is that it has sections dealing with stochastic differential equations and delay differential equations as well as ordinary differential equations and partial differential equations. Stochastic differential equations and delay differential equations are often studied only in advanced texts and courses; yet, the techniques used to analyze these equations are easy to understand and easy to apply. Had this book been available when I was a graduate student, it would have saved me much time. It has saved me time in solving problems that arose from my own work in industry (the Jet Propulsion Laboratory, Sandia Laboratories, EXXON Research and Engineering, The MITRE Corporation, BBN). Parts of the text have been utilized in differential equations classes at the Rensselaer Polytechnic Institute. Students’ comments have been used to clarify the text. Unfortunately, there may still be some errors in the text; I would greatly appreciate receiving notice of any such errors. Many people have been kind enough to send in suggestions for additional material to add and corrections of existing material. There are too many to name them individually, but Alain Moussiaux stands out for all of the checking he has performed. Thank you all! This book is dedicated to my wife, Janet Taylor. Boston, Mass. 1997 [email protected]

Daniel Zwillinger

c CD-ROM Handbook of Differential Equations Academic Press 1997

Introduction This book is a compilation of the most important and widely applicable methods for solving and approximating differential equations. As a reference book, it provides convenient access to these methods and contains examples of their use. The book is divided into four parts. The first part is a collection of transformations and general ideas about differential equations. This section of the book describes the techniques needed to determine whether a partial differential equation is well posed, what the “natural” boundary conditions are, and many other things. At the beginning of this section is a list of definitions for many of the terms that describe differential equations and their solutions. The second part of the book is a collection of exact analytical solution techniques for differential equations. The techniques are listed (nearly) alphabetically. First is a collection of techniques for ordinary differential equations, then a collection of techniques for partial differential equations. Those techniques that can be used for both ordinary differential equations and partial differential equations have a star (∗) next to the method name. For nearly every technique, the following are given: • • • • • • •

the types of equations to which the method is applicable the idea behind the method the procedure for carrying out the method at least one simple example of the method any cautions that should be exercised notes for more advanced users references to the literature for more discussion or more examples

The material for each method has deliberately been kept short to simplify use. Proofs have been intentionally omitted. It is hoped that, by working through the simple example(s) given, the method will be understood. Enough insight should be gained from working the example(s) to apply the method to other equations. Further references are given for each method so that the principle may be studied in more detail or so more examples may be seen. Note that not all of the references listed at the end of a method may be referred to in the text. The author has found that computer languages that perform symbolic manipulations (e.g., Macsyma, Maple, and Mathematica) are very useful for performing the calculations necessary to analyze differential equations. Hence, there is a section comparing the capabilities of these languages and, for some exact analytical techniques, examples of their use are given.

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Not all differential equations have exact analytical solutions; sometimes an approximate solution will have to do. Other times, an approximate solution may be more useful than an exact solution. For instance, an exact solution in terms of a slowly converging infinite series may be laborious to approximate numerically. The same problem may have a simple approximation that indicates some characteristic behavior or allows numerical values to be obtained. The third part of this book deals with approximate analytical solution techniques. For the methods in this part of the book, the format is similar to that used for the exact solution techniques. We classify a method as an approximate method if it gives some information about the solution but does not give the solution of the original equation(s) at all values of the independent variable(s). The methods in this section describe, for example, how to obtain perturbation expansions for the solutions to a differential equation. When an exact or an approximate solution technique cannot be found, it may be necessary to find the solution numerically. Other times, a numerical solution may convey more information than an exact or approximate analytical solution. The fourth part of this book is concerned with the most important methods for finding numerical solutions of common types of differential equations. Although there are many techniques available for numerically solving differential equations, this book has only tried to illustrate the main techniques for each class of problem. At the beginning of the fourth section is a brief introduction to the terms used in numerical methods. When possible, short Fortran or C programs1 have been given. Once again, those techniques that can be used for both ordinary differential equations and partial differential equations have a star next to the method name. This book is not designed to be read at one sitting. Rather, it should be consulted as needed. Occasionally we have used “ODE” to stand for “ordinary differential equation” and “PDE” to stand for “partial differential equation.” This book contains many references to other books. Whereas some books cover only one or two topics well, some books cover all their topics well. The following books are recommended as a first source for detailed understanding of the differential equation techniques they cover; each is broad in scope and easy to read.

References [1] Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers. McGraw–Hill Book Company, New York, 1978. [2] Boyce, W. E., and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, fourth ed. John Wiley & Sons, New York, 1986. [3] Butkov, E. Mathematical Physics. Addison–Wesley Publishing Co., Reading, MA, 1968. [4] Chester, C. R. Techniques in Partial Differential Equations. McGraw–Hill Book Company, New York, 1970. [5] Collatz, L. The Numerical Treatment of Differential Equations. Springer– Verlag, New York, 1966. 1 We make no warranties, express or implied, that these programs are free of error. The author and publisher disclaim all liability for direct or consequential damages resulting from your use of the programs.

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[6] Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice–Hall, Inc., Englewood Cliffs, NJ, 1971. [7] Ince, E. L. Ordinary Differential Equations. Dover Publications, Inc., New York, 1964. [8] Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis. Interscience Publishers, Inc., New York, 1958.

c CD-ROM Handbook of Differential Equations Academic Press 1997

Introduction to the Electronic Version This third edition of Handbook of Differential Equations is available both in print form and in electronic form. The electronic version can be used with any modern web browser (such as Netscape or Explorer). Some features of the electronic version include • Quickly finding a specific method for a differential equation Navigating through the electronic version is performed via lists of methods for differential equations. Facilities are supplied for creating lists of methods based on filters. For example, a list containing all the differential equation methods that have both a program and an example in the text can be created. Or, a list of differential equation methods that contain either a table or a specific word can be created. It is also possible to apply boolean operations to lists to create new lists. • Interactive programs demonstrating some of the numerical methods For some of the numerical methods, an interactive Java program is supplied. This program numerically solves the example problem described in the text. The parameters describing the numerical solution may be varied, and the resulting numerical approximation obtained. • Live links to the internet The third edition of this book has introduced links to relevant web sites on the internet. In the electronic version, these links are active (clicking on one of them will take you to that site). In the print version, the URLs may be found by looking in the index under the entry “URL.” • Dynamic rendering of mathematics All of the mathematics in the print version is available electronically, both through static gif files and via dynamic Java rendering.

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How to Use This Book This book has been designed to be easy to use when solving or approximating the solutions to differential equations. This introductory section outlines the procedure for using this book to analyze a given differential equation. First, determine whether the differential equation has been studied in the literature. A list of many such equations may be found in the “Look-Up” section beginning on page 179. If the equation you wish to analyze is contained on one of the lists in that section, then see the indicated reference. This technique is the single most useful technique in this book. Alternatively, if the differential equation that you wish to analyze does not appear on those lists or if the references do not yield the information you desire, then the analysis to be performed depends on the type of the differential equation. Before any other analysis is performed, it must be verified that the equation is well posed. This means that a solution of the differential equation(s) exists, is unique, and depends continuously on the “data.” See pages 15, 53, 101, and 115.

Given an Ordinary Differential Equation • It may be useful to transform the differential equation to a canonical form or to a form that appears in the “Look-Up” section. For some common transformations, see pages 128–162. • If the equation has a special form, then there may be a specialized solution technique that may work. See the techniques on pages 275, 278, and 398. • If the equation is a – – – – – –

Bernoulli equation, see page 235. Chaplygin equation, see page 511. Clairaut equation, see page 237. Euler equation, see page 281. Lagrange equation, see page 363. Riccati equation, see page 392.

• If the equation does not depend explicitly on the independent variable, see pages 230 and 411. • If the equation does not depend explicitly on the dependent variable (undifferentiated), see pages 260 and 409. xxi

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• If one solution of the equation is known, it may be possible to lower the order of the equation; see page 389. • If discontinuous terms are present, see page 264. • The single most powerful technique for solving analytically ordinary differential equations is through the use of Lie groups; see page 366.

Given a Partial Differential Equation Partial differential equations are treated in a different manner from ordinary differential equations; in particular, the type of the equation dictates the solution technique. First, determine the type of the partial differential equation; it may be hyperbolic, elliptic, parabolic, or of mixed type (see page 36). • It may be useful to transform the differential equation to a canonical form, or to a form that appears in the “Look-Up” Section. For transformations, see pages 146, 166, 168, 173, 456, and 467. • The simplest technique for working with partial differential equations, which does not always work, is to “freeze” all but one of the independent variables and then analyze the resulting partial differential equation or ordinary differential equation. Then the other variables may be added back in, one at a time. • If every term is linear in the dependent variable, then separation of variables may work; see page 487. • If the boundary of the domain must be determined as part of the problem, see the technique on page 311. • See all of the exact solution techniques, which are on pages 428–508. In addition, many of the techniques that can be used for ordinary differential equations are also applicable to partial differential equations. These techniques are indicated by a star with the method name. • If the equation is hyperbolic, – In principle, the differential equation may be solved using the method of characteristics; see page 432. Often, though, the calculations are impossible to perform analytically. – See the section on the exact solution to the wave equation on page 501. • The single most powerful technique for analytically solving partial differential equations is through the use of Lie groups; see page 471.

Given a System of Differential Equations • First, verify that the system of equations is consistent; see page 43. • Note that many of the methods for a single differential equation may be generalized to handle systems. c CD-ROM Handbook of Differential Equations Academic Press 1997

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• By using differential resultants, it may be possible to obtain a single equation; see page 50. • The following methods are for systems of equations: – The method of generating functions; see page 315. – The methods for constant coefficient differential equations; see pages 421 and 449. – The finding of integrable combinations; see page 334. • If the system is hyperbolic, then the method of characteristics will work (in principle); see page 432. • See also the method for Pfaffian equations (see page 384) and the method for matrix Riccati equations (see page 395).

Given a Stochastic Differential Equation • A general discussion of random differential equations may be found on page 91. • To determine the transition probability density, see the discussion of the Fokker–Planck equation on page 303. • To obtain the moments without solving the complete problem, see pages 568 and 572. • If the noise appearing in the differential equation is not “white noise,” the section on stochastic limit theorems might be useful (see page 629). • To numerically simulate the solutions of a stochastic differential equation, see the technique on page 775.

Given a Delay Equation See the techniques on page 253.

Looking for an Approximate Solution • If exact bounds on the solution are desired, see the methods on pages 545, 551, and 560. • If the solution has singularities that are to be recovered, see page 582. • If the differential equation(s) can be formulated as a contraction mapping, then approximations may be obtained in a natural way; see page 58.

Looking for a Numerical Solution • It is extremely important that the differential equation(s) be well posed before a numerical solution is attempted. See the theorem on page 723 for an indication of the problems that can arise. c CD-ROM Handbook of Differential Equations Academic Press 1997

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• The numerical solution technique must be stable if the numerical solution is to approximate the true solution of the differential equation; see pages 683, 688, and 692. • It is often easiest to use commercial software packages when looking for a numerical solution; see page 654. • If the problem is “stiff,” then a method for dealing with “stiff” problems will probably be required; see page 770. • If a low-accuracy solution is acceptable, then a Monte-Carlo solution technique may be used; see pages 810 and 844. • To determine a grid on which to approximate the solution numerically, see page 675. • To find an approximation scheme that works on a parallel computer, see page 755.

Other Things to Consider • • • • • • • • • • • •

Does the differential equation undergo bifurcations? See page 19. Is the solution bounded? See pages 551 and 560. Is the differential equation well posed? See pages 15 and 115. Does the equation exhibit symmetries? See pages 366 and 471. Is the system chaotic? See page 29. Are some terms in the equation discontinuous? See page 264. Are there generalized functions in the differential equation? See pages 318 and 330. Are fractional derivatives involved? See page 308. Does the equation involve a small parameter? See the perturbation methods (on pages 586, 590, 598, 605, 610, and 614) or pages 538, 642. Is the general form of the solution known? See page 415. Are there multiple time or space scales in the problem? See pages 538 and 605. Always check your results!

Methods Not Discussed in This Book There are a variety of novel methods for differential equations and their solutions not discussed in this book. These include 1. 2. 3. 4. 5. 6.

Adomian’s decomposition method (see Adomian [1]) Entropy methods (see Baker-Jarvis [2]) Fuzzy logic (see Leland [5]) Infinite systems of differential equations (see Steinberg [6]) Monodromy deformation (see Chowdhury and Naskar [3]) p-adic differential equations (see Dwork [4])

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References [1] Adomian, G. Stochastic Systems. Academic Press, New York, 1983. [2] Baker-Jarvis, J. Solution to boundary value problems using the method of maximum entropy. J. Math. and Physics 30, 2 (February 1989), 302–306. [3] Chowdhury, A. R., and Naskar, M. Monodromy deformation approach to nonlinear equations — A survey. Fortschr. Phys. 36, 12 (1988), 9399–953. [4] Dwork, B. Lectures on p-adic Differential Equations. Springer–Verlag, New York, 1982. [5] Leland, R. P. Fuzzy differential systems and Malliavin calculus. Fuzzy Sets and Systems 70 (1995), 59–73. [6] Steinberg, S. Infinite systems of ordinary differential equations with unbounded coefficients and moment problems. J. Math. Anal. Appl. 41 (1973), 685–694.

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Definitions and Concepts

Definition of Terms

Adiabatic invariant When the parameters of a physical system vary slowly under the effect of an external perturbation, some quantities are constant to any order of the variable describing the slow rate of change. Such a quantity is called an adiabatic invariant. This does not mean that these quantities are exactly constant but rather that their variation goes to zero faster than any power of the small parameter. Analytic A function is analytic at a point if the function has a power series expansion valid in some neighborhood of that point. Asymptotic equivalence Two functions, f (x) and g(x), are said to be asymptotically equivalent as x → x0 if f (x)/g(x) ∼ 1 as x → x0 , that is: f (x) = g(x) [1 + o(1)] as x → x0 . See Erd´elyi [4] for details. Asymptotic expansion Given a function f (x) and an asymptotic seP∞ a gk (x), where the {ak } are ries {gk (x)} at x0 , the formal series k k=0 given constants, is said to be an asymptotic expansion of f (x) if f (x) − Pn a g (x) = o(g (x)) as x → x for every n; this is expressed as f (x) ∼ k k n 0 Pk=0 ∞ a g (x). Partial sums of this formal series are called asymptotic k k k=0 approximations to f (x). Note that the formal series need not converge. See Erd´elyi [4] for details. Asymptotic series A sequence of functions, {gk (x)}, forms an asymptotic series at x0 if gk+1 (x) = o(gk (x)) as x → x0 . Autonomous An ordinary differential equation is autonomous if the independent variable does not appear explicitly in the equation. For example, yxxx + (yx )2 = y is autonomous while yx = x is not (see page 230). Bifurcation The solution of an equation is said to undergo a bifurcation if, at some critical value of a parameter, the number of solutions to the equation changes. For instance, in a quadratic equation with real coefficients, as the constant term changes the number of real solutions can change from 0 to 2 (see page 19). Boundary data Given a differential equation, the value of the dependent variable on the boundary may be given in many different ways. Dirichlet boundary conditions The dependent variable is prescribed on the boundary. This is also called a boundary condition of the first kind. Homogeneous boundary conditions The dependent variable vanishes on the boundary. Mixed boundary conditions A linear combination of the dependent variable and its normal derivative is given on the boundary, c CD-ROM Handbook of Differential Equations Academic Press 1997

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or one type of boundary data is given on one part of the boundary while another type of boundary data is given on a different part of the boundary. This is also called a boundary condition of the third kind. Neumann boundary conditions The normal derivative of the dependent variable is given on the boundary. This is also called a boundary condition of the second kind. Sometimes the boundary data also include values of the dependent variable at points interior to the boundary. Boundary layer A boundary layer is a small region, near a boundary, in which a function undergoes a large change (see page 590). Boundary value problem An ordinary differential equation, where not all of the data are given at one point, is a boundary value problem. For example, the equation y 00 + y = 0 with the data y(0) = 1, y(1) = 1 is a boundary value problem. Characteristics A hyperbolic partial differential equation can be decomposed into ordinary differential equations along curves known as characteristics. These characteristics are themselves determined to be the solutions of ordinary differential equations (see page 432). Cauchy problem The Cauchy problem is an initial value problem for a partial differential equation. For this type of problem there are initial conditions but no boundary conditions. Commutator If L[·] and H[·] are two differential operators, then the commutator of L[·] and H[·] is defined to be the differential operator given by [L, H] := L ◦ H − H ◦ L = −[H, L]. For example, the commutator of the d d operators L[·] = x dx and H[·] = 1 + dx is       d d d d d 1+ − 1+ x =− . [L, H] = x dx dx dx dx dx See Goldstein [6] for details. Complete A set of functions is said to be complete on an interval if any other function that satisfies appropriate boundedness and smoothness conditions can be expanded as a linear combination of the original functions. Usually the expansion is assumed to converge in the “mean square,” or L2 sense. For example, the functions {un (x)} := {sin(nπx), cos(nπx)} are complete on the interval [0, 1] because any C 1 [0, 1] function, f (x), can be written as ∞   X f (x) = a0 + an cos(nπx) + bn sin(nπx) n=1

for some set of {an , bn }. See Courant and Hilbert [3, pages 51–54] for details. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

Complete system The system of nonlinear partial differential equations: {Fk (x1 , . . . , xr , y, p1 , . . . , pr ) = 0 | k = 1, . . . , s}, in one dependent variable, y(x), where pi = dy/dxi , is called a complete system if each {Fj , Fk }, for 1 ≤ j, k ≤ r, is a linear combination of the {Fk }. Here { , } represents the Lagrange bracket. See Iyanaga and Kawada [8, page 1304]. Conservation form A hyperbolic partial differential equation is said to be in conservation form if each term is a derivative with respect to some variable. That is, it is an equation for u(x) = u(x1 , x2 , . . . , xn ) that has (u,x) the form ∂f1∂x + · · · + ∂fn∂x(u,x) = 0 (see page 47). 1 n Consistency

There are two types of consistency:

Genuine consistency This occurs when the exact solution to an equation can be shown to satisfy some approximations that have been made in order to simplify the equation’s analysis. Apparent consistency This occurs when the approximate solution to an equation can be shown to satisfy some approximations that have been made in order to simplify the equation’s analysis. When simplifying an equation to find an approximate solution, the derived solution must always show apparent consistency. Even then, the approximate solution may not be close to the exact solution, unless there is genuine consistency. See Lin and Segel [9, page 188]. Coupled systems of equations A set of differential equations is said to be coupled if there is more than one dependent variable and each equation involves more than one dependent variable. For example, the system {y 0 + v = 0, v 0 + y = 0} is a coupled system for {y(x), v(x)}. Degree The degree of an ordinary differential equation is the greatest number of times the dependent variable appears in any single term. For example, the degree of y 0 + (y 00 )2 y + 1 = 0 is 3, whereas the degree of y 00 y 0 y 2 + x5 y = 1 is 4. The degree of y 0 = sin y is infinite. If all the terms in a differential equation have the same degree, then the equation is called equidimensional-in-y (see page 278). Delay equation A delay equation, also called a differential delay equation, is an equation that depends on the “past” as well the “present.” For example, y 00 (t) = y(t − τ ) is a delay equation when τ > 0. See page 253. Determined A truncated system of differential equations is said to be determined if the inclusion of any higher order terms cannot affect the topological nature of the local behavior about the singularity. Differential form A first order differential equation is said to be in differential form if it is written P (x, y)dx + Q(x, y)dy = 0. Dirichlet problem The Dirichlet problem is a partial differential equation with Dirichlet data given on the boundaries. That is, the dependent variable is prescribed on the boundary. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Eigenvalues, eigenfunctions Given a linear operator L[·] with boundary conditions B[·], there will sometimes exist nontrivial solutions to the equation L[y] = λy (the solutions may or may not be required to also satisfy B[y] = 0). When such a solution exists, the value of λ is called an eigenvalue. Corresponding to the eigenvalue λ there will exist solutions {yλ (x)}; these are called eigenfunctions. See Stakgold [12, Chapter 7, pages 411–466] for details. Elliptic operator

The differential operator

n X i,j=1

aij

∂2 is an elliptic ∂xi ∂xj

differential operator if the quadratic form xT Ax, where A = (aij ), is positive definite whenever x 6= 0. If the {aij } are functions of some variable, say t, and the operator is elliptic for all values of t of interest, then the operator is called uniformly elliptic. See page 36. R Euler–Lagrange equation If u = u(x) and J[u] = f (u0 , u, x) dx, then the condition for the vanishing of the variational derivative of J with respect to u, δJ δu = 0 is given by the Euler–Lagrange equation:   d ∂ ∂ − f = 0. ∂u dx ∂u0 R If w = w(x) and J = g(w00 , w0 , w, x) dx, then the Euler–Lagrange equation is   d ∂ d2 ∂ ∂ − g = 0. + ∂w dx ∂w0 dx2 ∂w00 RR If v = v(x, y) and J = h(vx , vy , v, x, y) dx dy, then the Euler–Lagrange equation is   d ∂ d ∂ ∂ − h = 0. − ∂v dx ∂vx dy ∂vy See page 418 for more details. First integral: ODE When a given differential equation is of order n and, by a process of integration, an equation of order n − 1 involving an arbitrary constant is obtained, then this new equation is known as a first integral of the given equation. For example, the equation y 00 + y = 0 has the equation (y 0 )2 + y 2 = C as a first integral. First integral: PDE A function u(x, y, z) is called a first integral of dy dz the vector field V = (P, Q, R) (or of its associated system: dx P = Q = R) if at every point in the domain V is orthogonal to grad u, i.e., V ·∇u = P

∂u ∂u ∂u +Q +R = 0. ∂x ∂y ∂z

Conversely, any solution of this partial differential equation is a first integral of V. Note that if u(x, y, z) is a first integral of V, then so is f (u). c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

Fr´ echet derivative, Gˆ ateaux derivative The Gˆ ateaux derivative of the operator N [·], at the “point” u(x), is the linear operator defined by L[z(x)] = lim

→0

N [u + z] − N [u] . 

For example, if N [u] = u3 + u00 + (u0 )2 , then L[z] = 3u2 z + z 00 + 2u0 z 0 . If, in addition, ||N [u + h] − N [u] − L[u]h|| =0 lim ||h||→0 ||h|| (as is true in our example), then L[u] is also called the Fr´echet derivative of N [·]. See Olver [11] for details. Fuchsian equation A Fuchsian equation is an ordinary differential equation whose only singularities are regular singular points. Fundamental matrix The vector ordinary differential equation y0 = Ay for y(x), where A is a matrix, has the fundamental matrix Φ(x) if Φ satisfies Φ0 = AΦ and the determinant of Φ is nonvanishing (see page 119). General solution Given an nth order linear ordinary differential equation, the general solution contains all n linearly independent solutions, with a constant multiplying each one. For example, the differential equation y 00 + y = 1 has the general solution y(x) = 1 + A sin x + B cos x, where A and B are arbitrary constants. Green’s function A Green’s function is the solution of a linear differential equation, which has a delta function appearing either in the equation or in the boundary conditions (see page 318). Harmonic function equation: ∇2 φ = 0.

A function φ(x) is harmonic if it satisfies Laplace’s

Hodograph In a partial differential equation, if the independent variables and dependent variables are switched, then the space of independent variables is called the hodograph space (in two dimensions, the hodograph plane) (see page 456). Homogeneous equation

Used in two different senses:

• An equation is said to be homogeneous if all terms depend linearly on the dependent variable or its derivatives. For example, the equation yxx + xy = 0 is homogeneous whereas the equation yxx + xy = 1 is not. • A first order ordinary differential equation is said to be homogeneous if the forcing function is a ratio of homogeneous polynomials (see page 327).

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Ill posed problems A problem that is not well posed is said to be ill posed. Typical ill posed problems are the Cauchy problem for the Laplace equation, the initial/boundary value problem for the backward heat equation, and the Dirichlet problem for the wave equation (see page 115). Initial value problem An ordinary differential equation with all of the data given at one point is an initial value problem. For example, the equation y 00 + y = 0 with the data y(0) = 1, y 0 (0) = 1 is an initial value problem. Involutory transformation An involutory transformation T is one that, when applied twice, does not change the original system; i.e., T 2 is equal to the identity function. R∞ L2 function A function f (x) is said to belong to L2 if 0 |f (x)|2 dx is finite. Lagrange bracket If {Fj } and {Gj } are sets of functions of the independent variables {u, v, . . . } then the Lagrange bracket of u and v is defined to be  X  ∂Fj ∂Gj ∂Fj ∂Gj {u, v} = − = − {v, u} . ∂u ∂v ∂v ∂u j See Goldstein [6] for details. Lagrangian derivative The Lagrangian derivative (also called the ma∂F terial derivative) is defined by DF Dt := ∂t + v · ∇F , where v is a given vector. See Iyanaga and Kawada [8, page 669]. Laplacian The Laplacian is the differential operator usually denoted by ∇2 (in many books it is represented as ∆). It is defined by ∇2 φ = div(grad φ), when φ is a scalar. The vector Laplacian of a vector is the differential operator denoted by 4 5 (in most books it is represented as ∇2 ). It is defined by 4 5v = grad(div v) − curl curl v, when v is a vector. See Moon and Spencer [10] for details. Leibniz’s rule Leibniz’s rule states that ! Z g(t) Z g(t) ∂h d (t, ζ) dζ. h(t, ζ) dζ = g 0 (t)h(t, g(t)) − f 0 (t)h(t, f (t)) + dt f (t) f (t) ∂t Lie algebra A Lie algebra is a vector space equipped with a Lie bracket (often called a commutator) [x, y] that satisfies three axioms: • [x, y] is bilinear (i.e., linear in both x and y separately), • the Lie bracket is anti-commutative (i.e., [x, y] = −[y, x]), • the Jacobi identity, [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0, holds. See Olver [11] for details. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

Limit cycle A limit cycle is a solution to a differential equation that is a periodic oscillation of finite amplitude (see page 78). Linear differential equation A differential equation is said to be linear if the dependent variable appears only with an exponent of 0 or 1. For example, the equation x3 y 000 + y 0 + cos x = 0 is a linear equation, whereas the equation yy 0 = 1 is nonlinear. Linearize To linearize a nonlinear differential equation means to approximate the equation by a linear differential equation in some region. For example, in regions where |y| is “small,” the nonlinear ordinary differential equation y 00 + sin y = 0 could be linearized to y 00 + y = 0. Linearizable Partial differential equations that can be solved either by an appropriate inverse scattering scheme or by a transformation to a linear partial differential equation are said to be linearizable. Lipschitz condition If f (x, y) is a bounded continuous function in a domain D, then f (x, y) is said to satisfy a Lipschitz condition in y in D if |f (x, y1 ) − f (x, y2 )| ≤ Ky |y1 − y2 | for some finite constant Ky , independent of x, y1 , and y2 in D. If, for some finite constant Kx , f (x, y) satisfies |f (x1 , y) − f (x2 , y)| ≤ Kx |x1 − x2 | independent of x1 , x2 , and y in D, then f (x, y) satisfies a Lipschitz condition in x in D. If both of these conditions are satisfied and K = max(Kx , Ky ), then f (x, y) satisfies a Lipschitz condition in D, with Lipschitz constant K. This also extends to higher dimensions. See Coddington and Levinson [2] for details. Maximum principle There are many “maximum principles” in the literature. The most common is “a harmonic function attains its absolute maximum on the boundary” (see page 560). Mean value theorem This is a statement about the solutionRof Laplace’s R equation. It states, “If ∇2 u = 0 (in N dimensions), then u(z) = S u dS/ S dS where S is the boundary of a N -dimensional sphere centered at z.” For example, in N = 2, we have, “In 2 dimensions, the value of a solution to Laplace’s equation at a point is the average of the values on any circle about that point.” See Iyanaga and Kawada [8, page 624]. Metaparabolic equation A metaparabolic equation has the form L[u]+ M [ut ] = 0, where u = u(x, t), L[·] is a linear differential operator in x of degree n, M [·] is a linear differential operator in x of degree m, and m < n. If, conversely, m > n, then the equation is called pseudoparabolic. See Gilbert and Jensen [5] for details. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Natural Hamiltonian APnatural Hamiltonian is one having the form n H = T + V , where T = 12 k=1 p2k and V is a function of the position variables only (i.e., V = V (q) = V (q1 , . . . , qn )). Near identity transformation A near-identity transformation is a transformation in a differential equation from the old variables {a, b, c, . . . } to the new variables {α, β, γ, . . . } via a = α + A(α, β, γ, . . . ), b = β + B(α, β, γ, . . . ), c = γ + C(α, β, γ, . . . ), .. . where {A, B, C, . . . } are strictly nonlinear functions (i.e., there are no linear or constant terms). Very frequently {A, B, C, . . . } are taken to be homogeneous polynomials (of, say, degree N ) in the variables α, β, γ, . . . , with unknown coefficients. For example, in two variables we might take A(α, β) =

n X

Aj,n−j αj β n−j ,

B(α, β) =

j=0

n X

Bj,n−j αj β n−j ,

j=0

for some given value of n (see page 86). Neumann problem The Neumann problem is a partial differential equation with Neumann data given on the boundaries. That is, the normal derivative of the dependent variable is given on the boundary. See Iyanaga and Kawada [8, page 999]. Normal form An ordinary differential equation is said to be in normal form if it can be solved explicitly for the highest derivative; i.e., y (n) = G(x, y, y 0 , . . . , y (n−1) ). A system of partial differential equations (with dependent variables {u1 , u2 , . . . , um } and independent variables {x, y1 , y2 , . . . , yk }) is said to be in normal form if it has the form   ∂ r−1 um ∂u1 ∂u1 ∂ r um ∂ r uj , . . . , x, y , = F , . . . , y , u , . . . , u , , . . . , , . . . , j 1 k 1 m ∂xr ∂x ∂xr−1 ∂y1 ∂yk r for j = 1, 2, . . . , m. See page 86 or Iyanaga and Kawada [8, page 988]. Normal type An evolution equation is of normal type if it can be written in the form ut = un + h(u, u1 , . . . , um ) where n > m and uj = ∂ j u/∂xj . Nonlinear A differential equation that is not linear in the dependent variable is nonlinear. Nonoscillatory The real solution y(x) of yxx + f (x)y = 0 is said to be nonoscillatory in the wide sense in (0, ∞) if there exists a finite number c such that the solution has no zeros in [c, ∞]. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

Order of a differential equation The order of a differential equation is the greatest number of derivatives in any term in the differential equation. For example, the partial differential equation uxxxx = utt + u5 is of fourth order whereas the ordinary differential equation vx + x2 v 3 + v = 3 is of first order. Orthogonal Two vectors, x and y, are said to be orthogonal with respect to the matrix W if xT W y = 0 (often, W is taken to be the identity matrix). Two functions, say f (x) and g(x), are said toR be orthogonal with respect to a weighting function w(x) if (f (x), g(x)) := f (x)w(x)¯ g (x) dx = 0 over some appropriate range of integration. Here, an overbar indicates the complex conjugate. Oscillatory Consider the equation y 00 + f (x)y = 0 and the number of zeros it has in the interval [0, ∞]. If the number of zeros is infinite, then the equation (and the solutions) are called oscillatory. Pad´ e approximant A Pad´e approximant is a ratio of polynomials. The polynomials are usually chosen so that the Taylor series of the ratio is a prescribed function. See page 582. Particular solution Given a linear differential P equation, L[y] = f (x), the general solution can be written as y = yp + i Ci yi where yp , the particular solution, is any solution that satisfies L[y] = f (x). The yi are homogeneous solutions that satisfy L[y] = 0, and the {Ci } are arbitrary constants. If L[·] is an nth order differential operator, then there will be n linearly independent homogeneous solutions. Poisson bracket If f and g are functions of {pj , qj }, then the Poisson bracket of f and g is defined to be  X X  ∂f ∂g ∂f ∂g ∂(f, g) [f, g] = = − [g, f ] . = − ∂q ∂p ∂p ∂q ∂(q j j j j j , pj ) j j The Poisson bracket is invariant under a change of independent variables. See Goldstein [6] or Olver [11] for details. Quasilinear equation

Used in two different senses:

• A partial differential equation is said to be quasilinear in Pnif it is linear∂u = the first partial derivatives. That is, it has the form k=1 Ak (u, x) ∂x k B(u, x) when the dependent variable is u(x) = u(x1 , . . . , xn ) (see page 432). • A partial differential equation is said to be quasilinear if it has the form ut = g(u)ux(n) + f (u, ux, yx(2) , . . . , ux(n−1) ) for n ≥ 2. Radiation condition The radiation condition states that a wave equation has no waves incoming from an infinite distance, only outgoing waves. c CD-ROM Handbook of Differential Equations Academic Press 1997

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For example, the equation utt = ∇2 u might have the radiation condition u(x, t) ' A− exp(ik(t − x)) as x → −∞ and u(x, t) ' A+ exp(ik(t + x)) as x → +∞. This is also called the Sommerfeld radiation condition. See Butkov [1, page 617] for details. Riemann’s P function Riemann’s differential equation (see page 186) is the most general second order linear ordinary differential equation with three regular singular points. If these singular points are taken to be a, b, and c and the exponents of the singularities are taken to be α, α0 ; β, β 0 ; γ, γ 0 (where α + α0 + β + β 0 + γ + γ 0 = 1), then the solution to Riemann’s differential equation may written in the form of Riemann’s P function as   a b c y(x) = P  α β γ x . α0 β 0 γ 0 Robbins problem An elliptic partial differential equation with mixed boundary conditions is called a Robbins problem. See Iyanaga and Kawada [8, page 999]. Schwarzian derivative If y = y(x), then the Schwarzian derivative of y with respect to x is defined to be  2  2  00 0 1 y 00 y 000 3 y 00 y − = − . {y, x} ≡ y0 2 y0 y0 2 y0  2 dy If y = y(x) and z = z(x), then {z, x} = {z, y} dx + {y, x}. Therefore,  2 {y, x}. Note also that {y, x} is the unique elementary {x, y} = − dx dy function of the derivatives, whichnis invariant o under homographic transformations of x; that is, {y, x} = y, ax+b cx+d , where (a, b, c, d) are arbitrary constants with ad − bc = 1. See Ince [7, page 394]. Semi-Hamiltonian A diagonal system of equations having the form Ai (u)∂t ui = Bi (u)∂x ui is called semi-Hamiltonian if the coefficients satisfy Bi ∂ui Ak = Ai ∂ui Bk for i 6= k. Semilinear equations A partial differential equation is said to be semilinear if it has the form ut = ux(n) + f (u, ux, yx(2) , . . . , ux(n−1) ) for n ≥ 2. Shock A shock is a narrow region in which the dependent variable undergoes a large change. Also called a “layer” or a “propagating discontinuity.” See page 432. Singular point Given the homogeneous nth order linear ordinary differential equation y (n) + qn−1 (x)y (n−1) + qn−2 (x)y (n−2) + · · · + q0 (x)y = 0, the point x0 is classified as being an c CD-ROM Handbook of Differential Equations Academic Press 1997

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Ordinary point: if each of the {qi } are analytic at x = x0 . Singular point: if it is not an ordinary point. Regular singular point: if it is not an ordinary point and (x − x0 )i qi (x) is analytic for i = 0, 1, . . . , n. Irregular singular point: if it is not an ordinary point and not a regular singular point. The point at infinity is classified by changing variables to t = x−1 and then analyzing the point t = 0. See page 403. Singular solution A singular solution is a solution of a differential equation that is not derivable from the general solution by any choice of the arbitrary constants appearing in the general solution. Only nonlinear equations have singular solutions. See page 623. Stability The solution to a differential equation is said to be stable if small perturbations in the initial conditions, boundary conditions, or coefficients in the equation itself lead to “small” changes in the solution. There are many different types of stability that are useful. Stable A solution y(x) of the system y0 = f (y, x) that is defined for x > 0 is said to be stable if, given any  > 0, there exists a δ > 0 such that any solution w(x) of the system satisfying |w(0) − y(0)| < δ also satisfies |w(x) − y(x)| < . Asymptotic stability The solution u(x) is said to be asymptotically stable if, in addition to being stable, |w(x) − u(x)| → 0 as x → ∞. Relative stability The solution u(x) is said to be relatively stable if |w(0) − u(0)| < δ implies that |w(x) − u(x)| < u(x). See page 101 or Coddington and Levinson [2, Chapter 13] for details. Stefan problem A Stefan problem is one in which the boundary of the domain must be solved as part of the problem. For instance, when a jet of water leaves an orifice, not only must the fluid mechanics equations be solved in the stream, but the boundary of the stream must also be determined. Stefan problems are also called free boundary problems (see page 311). Superposition principle If u(x) and v(x) are solutions to a linear differential equation (ordinary or partial), then the superposition principle states that αu(x)+βv(x) is also a solution, where α and β are any constants (see page 413). Total differential equation A total differential equation is an equation P of the form: a (x) dxk = 0. See page 384. k k Trivial solution

The trivial solution is the identically zero solution.

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Turning points Given the equation y 00 + p(x)y = 0, points at which p(x) = 0 are called turning points. The asymptotic behavior of y(x) can change at these points. See page 645 or Wasow [13]. Weak solution A weak solution to a differential equation is a function that satisfies only an integral form of the defining equation. For example, a weak Rsolution of the differential equation a(x)y 00 − b(x) = 0 only needs to satisfy S [a(x)y 00 − b(x)] dx = 0 where S is some appropriate region. For this example, the weak solution may not be twice differentiable everywhere. See Zauderer [14, pages 288–294] for details. Well posed problems A problem is said to be well posed if a unique, stable solution that depends continuously on the data exists. See page 115. Wronskian Given the smooth functions {y1 , y2 , . . . , yn }, the Wronskian is the determinant y1 y2 ... yn 0 0 0 y1 y2 ... yn .. .. .. .. . . . . (n−1) (n−1) (n−1) y y . . . yn 1

2

If the Wronskian does not vanish in an interval, then the functions are linearly independent (see page 119).

References [1] Butkov, E. Mathematical Physics. Addison–Wesley Publishing Co., Reading, MA, 1968. [2] Coddington, E. A., and Levinson, N. Theory of Ordinary Differential Equations. McGraw–Hill Book Company, New York, 1955. [3] Courant, R., and Hilbert, D. Methods of Mathematical Physics. Interscience Publishers, Inc., New York, 1953. [4] Erd elyi, A. Asymptotic Expansions. Dover Publications, Inc., New York, 1956. [5] Gilbert, R. P., and Jensen, J. A computational approach for constructing singular solutions of one-dimensional pseudoparabolic and metaparabolic equations. SIAM J. Sci. Stat. Comput. 3, 1 (March 1982), 111–125. [6] Goldstein, H. Classical Mechanics. Addison–Wesley Publishing Co., Reading, MA, 1950. [7] Ince, E. L. Ordinary Differential Equations. Dover Publications, Inc., New York, 1964. [8] Iyanaga, S., and Kawada, Y. Encyclopedic Dictionary of Mathematics. MIT Press, Cambridge, MA, 1980. [9] Lin, C. C., and Segel, L. A. Mathematics Applied to Deterministic Problems in the Natural Sciences. The MacMillan Company, New York, 1974. c CD-ROM Handbook of Differential Equations Academic Press 1997

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[10] Moon, P., and Spencer, D. E. The meaning of the vector Laplacian. J. Franklin Institute 256 (1953), 551–558. [11] Olver, P. J. Applications of Lie Groups to Differential Equations. No. 107 in Graduate Texts in Mathematics. Springer–Verlag, New York, 1986. [12] Stakgold, I. Green’s Functions and Boundary Value Problems. John Wiley & Sons, New York, 1979. [13] Wasow, W. Linear Turning Point Theory, vol. 54. Springer–Verlag, New York, 1985. [14] Zauderer, E. Partial Differential Equations of Applied Mathematics. John Wiley & Sons, New York, 1983.

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Alternative Theorems

2.

15

Alternative Theorems

Applicable to

Linear ordinary differential equations.

Idea It is often possible to determine when a linear ordinary differential equation has a unique solution. Also, when the solution is not unique, it is sometimes possible to describe the degrees of freedom that make it non-unique.

Procedure Alternative theorems describe, in some way, the type of solutions to expect from linear differential equations. The most common alternative theorems for differential equations were derived by Fredholm. Suppose we wish to analyze the nth order linear inhomogeneous ordinary differential equation with boundary conditions L[u] = f (x), for i = 1, 2, . . . , n, Bi [u] = 0,

(2.1)

for u(x) on the interval x ∈ [a, b]. First, we must analyze the homogeneous equation and the adjoint homogeneous equation. That is, consider the two problems L[u] = 0, Bi [u] = 0,

for i = 1, 2, . . . , n,

L∗ [v] = 0, Bi∗ [v] = 0,

for i = 1, 2, . . . , n,

(2.2)

and (2.3)

where L∗ [·] is the adjoint of L[·], and the {Bi∗ [·]} are the adjoint boundary conditions (see page 95). Then Fredholm’s alternative theorem states that 1. If the system in (2.2) has only the trivial solution, that is u(x) ≡ 0, then (a) the system in (2.1) has a unique solution. (b) the system in (2.3) has only the trivial solution. 2. Conversely, if the system in (2.2) has k linearly independent solutions, say {u1 , u2 , . . . , uk }, then (a) the system in (2.3) has k linearly independent solutions, say {v1 , v2 , . . . , vk }. c CD-ROM Handbook of Differential Equations Academic Press 1997

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(b) the system in (2.1) has a solution if and only if the forcing function appearing in (2.1), f , is orthogonal to all solutions to Rb the adjoint system. That is (f, vi ) := a f (x)vi (x) dx = 0 for i = 1, 2, . . . , k. (c) the solution to (2.1), if 2(b) is satisfied, is given by u(x) = Pk u ¯(x) + j=i cj uj (x) for arbitrary constants {cj }, where u ¯(x) is any solution to (2.1).

Example 1 Given the ordinary differential equation for u(x) u0 + u = f (x), u(0) = 0,

(2.4)

we form the homogeneous system u0 + u = 0, u(0) = 0.

(2.5)

Because (2.5) has only the trivial solution, we know that the solution to equation (2.4) is unique. By the method of integrating factors (see page Rx 356), the solution to (2.4) is found to be u(x) = 0 f (t)et−x dt.

Example 2 Given the ordinary differential equation for u(x) u0 + u = f (x), u(0) − eu(1) = 0,

(2.6)

we form the homogeneous system u0 + u = 0, u(0) − eu(1) = 0.

(2.7)

In this case, (2.7) has the single non-trivial solution u(x) = e−x . Hence, the solution to (2.6) is not unique. To find out what restrictions must be placed on f (x) for (2.6) to have a solution, consider the corresponding adjoint homogeneous equation v 0 − v = 0, −ev(0) + v(1) = 0.

(2.8)

Since (2.8) has a single non-trivial solution, v(x) = ex , we conclude that (2.6) has a solution if and only if Z 1 f (t)et dt = 0. (2.9) 0

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If equation (2.9) is satisfied, then the solution of (2.6) will be given by Z x u(x) = Ce−x + f (t)et−x dt 0

where C is an arbitrary constant.

Example 3

The solution(s) to xy 00 − (1 + x)y 0 + y = 0 depends on the boundary conditions as follows: 1. With y(1) = 1, y 0 (1) = 2, the solution is y = 3ex−1 − (1 + x). 2. With y(0) = 1, y 0 (0) = 2, there is no solution. 3. With y(0) = 1, y 0 (0) = 1, there are infinitely many solutions of the form y = C(ex − 1 − x) + 1 + x.

Notes 1. Epstein [1, pages 83 and 111] discusses the Fredholm theorems in the general setting of a Banach space and a Hilbert space. 2. Interpretation of alternative theorems is usually straightforward when the underlying physics are understood. For example, the system −u00 = f (x),

0<x<1 u0 (0) = a1 , −u0 (1) = a2 R1 must satisfy the relation 0 f (x) dx = a1 + a2 . This states that for a rod experiencing one-dimensional heat flow, a steady state is possible only if the heat supplied along the rod is removed at the ends. 3. A generalized Green’s function is a Green’s function (see page 318) for a differential equation that does not have a unique solution. See Greenberg [2] for more details. 4. The Sturm–Liouville problem for u(x) on the interval x1 ≤ x ≤ x2   du d p(x) + q(x)u = f (x) (2.10) − dx dx −p(x1 )u0 (x1 ) + r1 u(x1 ) = 0

p(x2 )u0 (x2 ) + r2 u(x2 ) = 0

can be written as Z x2 h i 2 p(t)u0 (t) + q(t)u2 (t) dt + r1 u2 (x1 ) + r2 u2 (x2 ) x1 Z x2 f (t)u(t) dt + g1 u(x1 ) + g2 u(x2 ). = x1

Hence, if p(x) is positive, q(x), r1 , and r2 are non-negative and if R x2 f (t)u(t) dt+g1 u(x1 )+g2 u(x2 ) = 0, then there is a unique solution x1 to (2.10). 5. See also Haberman [3, pages 307–314] and Stakgold [4, pages 82–90, 207–214, and 319–323].

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References [1] Epstein, B. Partial Differential Equations: An Introduction. McGraw–Hill Book Company, New York, 1962. [2] Greenberg, M. D. Application of Green’s Functions in Science and Engineering. Prentice–Hall, Inc., Englewood Cliffs, NJ, 1971. [3] Haberman, R. Elementary Applied Partial Differential Equations. Prentice– Hall, Inc., Englewood Cliffs, NJ, 1968. [4] Stakgold, I. Green’s Functions and Boundary Value Problems. John Wiley & Sons, New York, 1979.

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Bifurcation Theory

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Bifurcation Theory

Applicable to

Nonlinear differential equations.

Idea Given a nonlinear differential equation that depends on a set of parameters, the number of distinct solutions may change as the parameters change. Points where the number of solutions change are called bifurcation points.

Procedure Although bifurcations occur in all types of equations, we restrict our discussion to ordinary differential equations. Consider the autonomous system dx = f(x; α), dt

(3.1)

where x and f are n-dimensional vectors and α is a set of parameters. Define the Jacobian matrix by   df ∂fi = J(x; α) := (x; α) | i, j = 1, . . . , n . (3.2) dx ∂xj Note that J(x; α)z is the Fr´echet derivative of f, at the point x (see page 6). Using the solution x(t, α) of equation (3.1), the values of α where one or more of the eigenvalues of J are zero are defined to be bifurcation points. At such points, the number of solutions to equation (3.1) may change, and the stability of the solutions might also change. If any of the eigenvalues have positive real parts, then the corresponding solution is unstable. If we are concerned only with the steady-state solutions of equation (3.1), as is often the case, then the bifurcation points will satisfy the simultaneous equations f(x; α) = 0,

and

det J = 0.

(3.3)

Define the eigenvalues of the Jacobian matrix defined in equation (3.2) to be {λi | i = 1, . . . , n}. We now presume that equation (3.1) depends on the single parameter α. Suppose that the change in stability is at the point α = α b, where the real part of a complex conjugate pair of eigenvalues (λ1 = λ2 ) pass through zero: α) = 0, <λ1 (b

=λ1 (b α) > 0,

<λ01 (b α) = 6 0,

and, for all values of α near α b, <λi (α) < 0 for i = 3, . . . , n. Then, under certain smoothness conditions, it can be shown that a small amplitude periodic solution exists for α near α b. Let  measure the c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

!

.. . ... . .. .... .. .. .. .. . ... ... .. ... .. . . .. . . .. ... .. .. ... .. ... .. .. ... .. .. .... .. .. . . .. .. .. .... .. ... ... ... ... ... ... ... ... ... .... ... ... . . ... .. .... ... .... .... .... .... .... .... .... . ..... ..... .......... ..... ..................... . . . . . .. ............................. ....... ...................... ...... ....... ...... ...... ...... ........ ....... ......... ......... . ............... . . . . . . . . . . . . ............................................

R



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... .... . .... ....

g

Figure 3.1: A bead on a spinning semi-circular wire. amplitude of the periodic solution. Then there are functions µ() and τ (), defined for all sufficiently small, real , such that µ(0) = τ (0) = 0 and that the system with α = α b + µ() has a unique small amplitude α). When expanded, we have solution of period T = 2π (1 + τ ()) /=λ1 (b µ() = µ2 2 + O(3 ). The sign of µ2 indicates where the oscillations occur, i.e., for α < α b or for α > α b.

Example 1 The nonlinear ordinary differential equation du = g(u) = u2 − λ1 u − λ2 dt

(3.4)

has steady-state solutions that satisfy g(u) = u2 − λ1 u − λ2 = 0. These steady-state solutions have bifurcation points given by dg = 2u − λ1 = 0. du Solving these last two equations simultaneously, it can be shown that the bifurcation points of the steady-state solutions are along the curve 4λ2 + λ21 = 0. Further analysis shows that equation (3.4) will have two real steady-state solutions when 4λ2 + λ21 > 0, and it will have no real steadystate solutions when 4λ2 + λ21 < 0.

Example 2 Consider a frictionless bead that is free to slide on a semi-circular hoop of wire of radius R that is spinning at an angular rate ω (see figure 3.1). The equation for θ(t), the angle of the bead from the vertical, is given by d2 θ g sin θ + dt2 R

  ω2R 1− cos θ = 0, g

(3.5)

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where g is the magnitude of the gravitational force. We define the parameter ν by ν = g/ω 2 R. We will analyze only the case ν ≥ 0. The three possible steady solutions of equation (3.5) are given by for ν ≥ 0, θ(t) = θ1 = 0, for ν ≤ 1, θ(t) = θ2 = cos−1 ν, for ν ≤ 1, θ(t) = θ3 = − cos−1 ν. Therefore, for ν > 1 (which corresponds to slow rotation speeds), the only steady solution is θ(t) = θ1 . For ν ≤ 1, however, there are three possible solutions. The solution θ(t) = θ1 will be shown to be unstable for ν < 1. To determine which solution is stable in a region where there are multiple solutions, a stability analysis must be performed. This is accomplished by assuming that the true solution is slightly perturbed from the given solution, and the rate of change of the perturbation is obtained. If the perturbation grows, then the solution is unstable. Conversely, if the perturbation decays (stays bounded), then the solution is stable (neutrally stable). First we perform a stability analysis for the solution θ(t) = θ1 . Define θ(t) = θ1 + φ(t),

(3.6)

where  is a small number and φ(t) is an unknown function. Using (3.6) in equation (3.5), and expanding all terms for   1, results in ν−1 d2 φ φ = O(). +g dt2 ν

(3.7)

The leading order terms in equation (3.7) represent the Fr´echet derivative of equation (3.5) at the “point” θ(t) = θ1 , applied to the function φ(t). The solution of this differential equation for φ(t), to leading order in , is φ(t) = A cos αt + B sin αt, (3.8) q
. If ν > 1, then where A and B are arbitrary constants and α = g ν−1 ν α is real, and the solutions for φ(t) remain bounded. Conversely, if ν < 1 then α becomes imaginary, and the solution in (3.8) becomes unbounded as t increases. Hence, the solution θ(t) = θ1 is unstable for ν < 1. Now we perform a stability analysis for the solution θ(t) = θ2 . Writing θ(t) = θ2 + ψ(t) and using this form in equation (3.5) leads to the equation for ψ(t): 1 − ν2 d2 ψ ψ = O(). +g 2 dt ν

(3.9)

The leading order terms in equation (3.9) represent the Fr´echet derivative of equation (3.5) at the “point” θ(t) = θ2 , applied to the function ψ(t). The c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

. ..... ....... ... ... ... . ... ............. .. ............................. . ................. .................. ... ............... ................. ... ................. ... ............. ............... ... .............. ........... ... ............ ........... ... .......... ........ ... ....... ..... ... .... ... ... ...... .. ... ... ..................................................... ... ......... . .. ....... ... .... . . . .. . . ...... . . . . . ... . ........ ... ........ ........... .. ............. . ........... ... .............. .............. . . . . . . . . . ... . . . ....... ................ .. ................. . ................. ... ............... .................. ... ................... .................. .. .. .. .. . .. .

S

U S

S

1



Figure 3.2: Bifurcation diagram for equation 3.6. A branch with the label “S” (“U”) is a stable (unstable) branch. solution of this differential equation for ψ(t) is ψ(t) βt + B sin βt, q = A cos
2 . If ν < 1, then where A and B are arbitrary constants and β = g 1−ν ν β is real and the solutions for ψ(t) remain bounded. Therefore, the solution θ(t) = θ2 is stable for ν < 1. In an exactly analogous manner, θ(t) = θ3 is stable for ν < 1. From what we have found, we can construct the bifurcation diagram shown in figure 3.2. In this diagram, the unstable steady solutions are indicated by a dashed line and the letter “U”, and the stable steady solutions are indicated by the solid line and the letter “S”. In words, this diagram states: • For no rotation (ω = 0 or ν = ∞), the only solution is θ(t) = θ1 = 0. • As the frequency of rotation increases (and so ν decreases), the solution θ(t) = θ1 becomes unstable at the bifurcation point ν = 1. • For ν < 1, the are two stable solutions, θ(t) = θ2 and θ(t) = θ3 . In this example, there is no way to know in advance which of these two solutions will occur (physically, the bead can slide up either side of the wire). The formula in (3.3) can be applied to equation (3.5) to determine the location of the bifurcation point without performing all of the above analysis. If we define x1 = θ and x2 = dθ dt , then equation (3.5) can be written as the system of ordinary differential equations     d x1 x2
= f(x) = , −g sin x1 1 − cosνx1 dt x2 which has the Jacobian matrix  df = J= g cos x1 + dx

g ν

 0 1
. cos2 x1 − sin2 x1 0

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If ν > 1, then no choice of (x1 , x2 ) will allow both f and det J to be zero simultaneously. For ν = 1, however, x1 = x2 = 0 make both f and det J equal to zero. Hence, a bifurcation occurs at ν = 1.

Example 3 Abelson [1] has developed a computer program in LISP that automatically explores the steady-state orbits of one-parameter families of periodically driven oscillators. The program generates both textual descriptions and schematic diagrams. For example, consider Duffing’s equation in the form x ¨ + 0.1x˙ + x3 = p cos t, where the parameter p is in the range [1, 25] and only those solutions with −5 ≤ x˙ ≤ 5 and −10 ≤ x ¨ ≤ 10 are considered. The program produced the graphical output shown in figure 3.3, along with the following textual description: The system was explored for values of p between 1 and 25, and 10 classes of stable periodic orbits were identified. Class A is already present at the start of the parameter range p = 1 with a family of order-1 orbits A0 . Near p = 2.287, there is a supercritical-pitchfork bifurcation, and A0 splits into symmetric families A1,0 and A1,1 , each of order 1. A1,0 vanishes at a fold bifurcation near p = 3.567. A1,1 vanishes similarly. Class B appears around p = 3.085 with a family of order-1 orbits B0 arising from a fold bifurcation. As the parameter p increases, B0 undergoes a period doubling cascade, reaching order 2 near p = 4.876, and order 4 near p = 5.441. Although the cascade was not traced past the order 4 orbit, there is apparently another period-doubling near p = 5.52, and a chaotic orbit was observed at p = 5.688. .. . Class J appears around p = 23.96 as a family of order-5 orbits J0 arising from a fold bifurcation. J0 is present at the end of the parameter range at p = 25. This program is capable of recognizing the following types of bifurcations: fold bifurcations, supercritical and subcritical flip bifurcations, supercritical and subcritical Niemark bifurcations, supercritical and subcritical pitchfork bifurcations, and transcritical bifurcations.

Notes 1. There are many different types of bifurcations. See figure 3.4 for diagrams of some of the following bifurcations: c CD-ROM Handbook of Differential Equations Academic Press 1997

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Figure 3.3: Graphical output generated automatically from the Bifurcation Interpreter in Abelson [1]. For Duffing’s equation, the evolution of 10 classes of families of periodic orbits and their bifurcations has been traced. The p values along the horizontal axis indicate the parameter value at which the bifurcations occur. (Reprinted with permission from Comp & Maths. With Appls. 20, 8, Abelson, H., The bifurcation interpreter: A step towards the automatic analysis of dynamical systems, Copyright 1990, Pergamon Press.) • Hopf bifurcation: a stable steady solution bifurcates into a stable oscillatory solution. That is, there are no stable steady solutions in that particular region of parameter space. This occurs by having some of the eigenvalues of the Jacobian in (3.2) become c CD-ROM Handbook of Differential Equations Academic Press 1997

3.

Bifurcation Theory . . ...... . .. . . . . . ....... . . ...... ...... . . ...... . .... . . .... ... .. ... ................................. .. . ... .. ... . . . . . . . . .. . . . . . ..... . . . .

.. ..... .. ..... .. ..... . .. ...... .... .. ...... ...... ...... ...... ...... ... .......... ...... . ..... ... .......... ... ................................. ...... ... ..... ... ..... ..... . ..... ..... .. ...... . . .. . ..... .. ..

fold

transcritical

25 . . ...... . .. . . . . . . ..... . ....... . . ........... . ......... . . . ... . . .. ................................. . . . .... ... . ........ . . . . . . ....... . ...... . ...... . . . . .

supercritical pitchfork

. . ...... . .. . . . . . ..... . . .... ..... . . . .... . . ... ... . ................................. .. ... ... .. . ... . . . . . . . .. . . . . . ..... . . . .

subcritical pitchfork

Figure 3.4: Diagrams of some types of bifurcations. Unstable solutions are indicated by dashed lines; stable solutions are indicated by solid lines.

purely imaginary. • Fold bifurcation: on one side of the bifurcation point a stable and an unstable periodic point (of the same order) coexist. On the other side of the bifurcation point, both periodic points have vanished. • Flip bifurcation (supercritical): a stable periodic point of order n transitions to a stable periodic point of order 2n and an unstable periodic point of order n. • Flip bifurcation (subcritical): an unstable periodic point of order 2n and a stable periodic point of order n transition to an unstable periodic point of order n. • Niemark bifurcation (supercritical): a stable periodic transitions to an unstable periodic point and a stable limit cycle. • Niemark bifurcation (subcritical): a stable periodic point and unstable limit cycle transition to an unstable periodic point. • Pitchfork bifurcation (supercritical): a stable periodic point transitions to two stable periodic points and an unstable periodic point, all of the same order. • Pitchfork bifurcation (subcritical): a stable periodic point and two unstable periodic points transition to an unstable periodic point. • Transcritical bifurcation: a stable periodic point and an unstable periodic point exchange stabilities; on the other side of the bifurcation point, the extrapolated stable point is now unstable, and vice-versa. 2. For a differential equation that is not autonomous, bifurcations can also occur from time-dependent solutions to other time-dependent solutions. 3. For the general finite dimensional mapping, G(x), from Rm to Rn , the Jacobian J(x) := ∂G ∂x need not be square. In this case, the critical points (which include the bifurcation points) are in the set C, with C := {x | x ∈ Rm , rank J(x) < min(m, n)} . c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

The regular points are Rm − C. The critical values are the values in the set G(C) := {y | y ∈ Rn , y = G(x) for some x ∈ C}. The regular values are Rn − G(C). 4. Sacks [8] describes the program POINCARE, which classifies bifurcation points and constructs representative phase diagrams for each type of behavior. The program is available directly from Sacks. 5. Numerical methods for computing bifurcations are described in Guckenheimer et al. [3] and Jepson and Spence [6].

References [1] Abelson, H. The bifurcation interpreter: A step towards the automatic analysis of dynamical systems. Comp. & Maths. with Appls. 20, 8 (1990), 13–35. [2] Guckenheimer, J. Patterns of bifurcations. In New Approaches to Nonlinear Problems in Dynamics, P. J. Holmes, Ed. SIAM, Philadelphia, PA, 1980, pp. 71–104. [3] Guckenheimer, J., Myers, M., and Sturmfels, B. Computing Hopf bifurcations. SIAM J. Numer. Anal. 34, 1 (February 1997), 1–21. [4] Holodniok, M., and Kubcek, M. New algorithms for the evaluation of complex bifurcation points in ordinary differential equations. A comparative numerical study. Appl. Math. and Comp. 15 (1984), 261–274. [5] Iooss, G., and Joseph, D. D. Elementary Stability and Bifurcation Theory. Springer–Verlag, New York, 1989. [6] Jepson, A. D., and Spence, A. Numerical methods for bifurcation problems. In The State of the Art in Numerical Analysis, A. Iserles and M. J. D. Powell, Eds. Clarendon Press, Oxford, England, 1987, pp. 273–298. [7] Rand, R. H., and Armbruster, D. Perturbation Methods, Bifurcation Theory and Computer Algebra. No. 65 in Applied Mathematical Sciences. Springer–Verlag, New York, 1987. [8] Sacks, E. Automatic analysis of one-parameter planar ordinary differential equations by intelligent numeric simulation. Artificial Intelligence 48 (1991), 27–56. [9] Seydel, R. From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis. American Elsevier Publishing Company, New York, 1988. [10] Wood, E. F., Kempf, J. A., and Mehra, R. K. Bistab: A portable bifurcation and stability analysis package. Appl. Math. and Comp. 15 (1984), 345–355.

c CD-ROM Handbook of Differential Equations Academic Press 1997

4.

A Caveat for Partial Differential Equations

4.

27

A Caveat for Partial Differential Equations

Idea To solve partial differential equations correctly, a good understanding of the nature of the partial differential equation is required. This requires more than a knowledge of the “physics” of the problem: a thorough understanding of the type of partial differential equation is needed. From Collatz [1, page 260]: That an investigation of the situation is absolutely essential is revealed even by quite simple examples; they show that formal calculation applied to partial differential equations can lead to false results very easily and that approximate methods can converge in a disarmingly innocuous manner to values bearing no relation to the correct solution.

Example Suppose we wish to solve the following wave equation (this example is from Collatz [1]) uxx = utt , u(x, 0) = cos x, for |x| < π/2, ∂u(x, 0) = cos x, for |x| < π/2,  ∂t  π u ± , t = sin t, for t > 0. 2

(4.1.a-d)

We will attempt to solve (4.1) by looking for a series solution of the form ∞ X

u(x, t) =

amn xm tn .

(4.2)

n,m=0

Using (4.2) in (4.1.a), we find that am,n+2 =

(m + 2)(m + 1) am+2,n . (n + 2)(n + 1)

(4.3)

To satisfy (4.1.b), we require ak,1 = 0. To satisfy (4.1.c), we also require ( 0, k = odd, (4.4) ak,0 = (−1)q /(2q)!, k = even = 2q. c CD-ROM Handbook of Differential Equations Academic Press 1997

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I.A

Definitions and Concepts

. ...... ....... .. .. ... .. ... .. .. ... ... ... ....... ..... .......... ............... . . . ..... . .. . ..... ..... . .. . .... .... . . .. . . .... .... . . .. . . .... ...... . . .. . . ...... ...... . . . . . ... . . . . . ......... . . . . ... . .. .... . . . . . . . . . .... ..... . . . . .. . . . . ..... ...... . . . . .. . . . . ...... ..... . . . . . .... . . . . . ........ . . . . . . . . . .. .. . . . . . ..... . . . . . . .. . . . . . . ..... ..... . . . . . . ... . . . . . . ......... .... ... . . . . . . . . .. . . . . . . . . .... .... . . . . . . . . .. . . . . . . . . ..... ...... . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . ........ . . . .... . . . . . . . . . .. . . . . . . . . . ..... ...... . . . . . . . . . .. . . . . . . . . . ...... .... ..... . . . . . . . . . . ... . . . . . . . . . . ...... .. .................................................................................................................................................................................................................................................... .. ........ .. ...... ...

t  2

,

solution valid here

2

x 2

Figure 4.1: Depiction of the characteristics and the range of validity of the solution found for equation 4.1. Evaluating equation (4.2) at x = 0 and using equations (4.3) and (4.4), we find that u(0, t) =

∞ X k=0

a0,k tk =

∞ X (−1)q q=0

(2q)!

t2q = cos t.

(4.5)

Now the conclusion in equation (4.5) is correct but only for 0 ≤ t ≤ π/2. This is because the characteristics (see page 432), t = π/2 ± x, emanating from the points (π/2, 0) and (−π/2, 0) do not allow u(0, t) to be determined directly for t > π/2. See figure 4.1 for a graphical representation of the characteristics of (4.1) and the region of validity for the solution in (4.5).

References [1] Collatz, L. The Numerical Treatment of Differential Equations. Springer– Verlag, New York, 1966. [2] Rassias, J. M. Counter Examples in Differential Equations and Related Topics. World Scientific, Singapore, 1991.

c CD-ROM Handbook of Differential Equations Academic Press 1997

5.

Chaos in Dynamical Systems

5.

29

Chaos in Dynamical Systems

Applicable to

Nonlinear differential equations.

Yields Information on whether or not a system is chaotic.

Idea Chaos is a phenomenon that can appear in solutions to nonlinear differential equations. Chaos is easily defined and can be easily (numerically) found in some equations.

Procedure For simplicity, we focus on deterministic systems modeled by coupled, autonomous, first order, ordinary differential equations of the form dxi = gi (x; q) for i = 1, 2, . . . , n dt

(5.1)

where x = (x1 , x2 , . . . , xn ) is the state-space vector and q = (q1 , q2 , . . . , qm ) is a set of parameters. This equation determines a set of solutions, each specified by their initial values. We can specify the solution corresponding to the initial condition p by x(t; p). Consider a set of initial conditions contained in a vanishing small volume V . Under the action of equation (5.1), the volume will change as a function of t. Precisely, ! Z Z X m dV ∂gi = ··· dx1 · · · dxn . dt ∂xi i=1 V

The summation term is the generalized divergence of g and is called the Lie derivative. Dissipative systems are characterized by contracting volumes; this is equivalent to dV /dt < 0. Conservative or Hamiltonian systems, in which equation (5.1) are Hamilton’s equations, obey Liouville’s theorem: dV /dt = 0. Any trajectory of a dissipative system as t → ∞ will approach a bounded region of phase space called an attractor. An attractor has zero volume in phase space. Attractors include points, limit cycles, and tori. For example, consider an unforced damped pendulum. The attractor for this is a point in phase space, the stable configuration with the pendulum hanging straight down. In this case, starting the pendulum swinging with slightly different initial conditions will lead to close paths in phase space and the same final state. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

For nonlinear systems exhibiting chaos, the separation of two nearby trajectories increases exponentially with time. This is referred to as sensitive dependence on initial conditions. For dissipative systems, a stretching in one direction has to be accompanied by a more-than-compensating contraction in other directions, so that the volume of an arbitrary droplet of initial conditions will contract with time. The phase-space trajectories for a chaotic system asymptotically approach a strange attractor, an attractor with a fractional dimension (i.e., a fractal). Lyapunov exponents are a measure of the rate of divergence (or convergence) of initially infinitesimally separated trajectories. The ith Lyapunov exponent, λi , can be found by considering the evolution of a vanishingly small set of initial conditions that form a hyperellipsoid. We define    1 ρi (t) λi := t→∞ (5.2) lim t ρi (0) ρ (0)→0 i

where ρi (t) is the length of the ith principal axis of the hyperellipsoid at time t, for i = 1, 2, . . . , n. An attractor is chaotic if it has at least one positive Lyapunov exponent. The Lyapunov exponents can be determined by analyzing the linearized equations corresponding to equation (5.1). For illustrative purposes, we specialize to n = 3 for the rest of this section. Consider the two close initial points: p0 = (x0 , y0 , z0 ) and p1 = p0 + δx = (x0 + δx, y0 + δy, z0 + δz). We want to find the evolution of the difference a(t) := x(t; p1 ) − x(t; p0 ). Using Taylor series d[x1 (t; p1 ) − x1 (t; p0 )] d[g1 (x(t; p0 + δx)) − g1 (x(t; p0 ))] da1 = = dt dt dt ∂g1 ∂g1 ∂g1 δx + δy + δz ≈ ∂x ∂y ∂z ∂g1 ∂g1 ∂g1 a1 + a2 + a3 , = ∂x ∂y ∂z where the partial derivatives are evaluated at x(t; p0 ). In general  ∂g  ∂g ∂g 1

1

∂x da  2 = M (x)a =  ∂g ∂x dt ∂g3

∂y ∂g2 ∂y ∂g3 ∂y

∂x

1

∂z ∂g2  ∂z  a, ∂g3 ∂z

where M is the Jacobian of the vector g. The Lyapunov exponents are related to the eigenvalues of the matrix M . In special situations, analytical methods can be used to obtain the Lyapunov spectra, while numerical methods must be used in general. When there is a stationary solution given by dx dt = g(x) = 0, the Jacobian matrix is time independent, and we can analytically obtain the (possibly complex) c CD-ROM Handbook of Differential Equations Academic Press 1997

5.

Chaos in Dynamical Systems x 1

. .. .. .... .... .... .... .... ..... ..... .... .... .... ..... ..... .... ..... ....... ..... . . ...... ... .. .. .. .. .. .. .. ... ... .. .. .. ... . . .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. . .. . . .... ... .... .... .... ... .... .... ... ... ... .... ... ... .... . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... .. ... ... .. .. ... .. .. .. .. ... . . . . .. . . . . .. .. . .. .. . . .. .. . .. ... . .. ... .... ... .... ... ... ... ... ... ... ... ... ... ... ... ... .. . ... ... .. .. ... .... ... ... ... .... ... .... ... .... ... ... .... ... .... .... ... .. .. .... ..... ..... .... ..... ..... ..... ...... ....... .... ... . .. . .. .. .. .

0

31 y 0.5 0.0

50

75

t

100

... ..... .... ..... ... . . . ....... .... . ...... ..... ... .. . .. .. . ...... ...... . ...... ..... . . .... ... . . ..... .. . ... .... . . . ... . . ... . ... . . .. . .. . . . . . .. . ...... ... . . .. ...... ................................................................................................................... .. ................................................... . . . .. . . .. ... . . . . ... . ... . . . ... . .. . . ... . . . ..... .. . . ..... ... . ....... . ... . ......... .... . ....................... . . . . . . . . . . . .

1

x

Figure 5.1: Duffing equation with Γ = 0.20. (Period 1 solution.) x 1

.. . ... .... ... ... .. .... ..... ...... ..... ...... .... .... .... .... . .. ... ..... .... ........ ..... .. .. ..... .. .... .. .. . .... .. .. . . . .. .. .. .. ... .. .. .. ... .. ... .. ... .. ... .. . .. .. .. . ... .... .... .... .. ... ... .... .. .... .. .... .... .... .. .... . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . ... ... .... .... .... ... .... .... ... ... ... .... ... .... .... ... ... .. . .. . .. . . .. .. .. . .. .. .. .. .. .. . .. .. .. .. .. . . ... ... .... ... ....... .... ... ...... .... .. .... ... .... ... ...... .. .. ... ... . .. ... .... .. ... .... .... .. ... ... .. ... .... .. .. ... ... .. .. ... ... ... ... . . .. .. .. .. .. .. .... ..... .... .... .... ... ..

0

y 0.5 0.0

50

75

t

100

... ...... .... . ... . . .. .... . ..... ..... ...... .. .. .. .......... .. ... .... ... . .... ...... .. . .... ....... ....... . .. ....... . ....... ....... ....... . . ...... ......... ...... . ..... . . . . . . . . ...... . . . ... .... ..... . . . . . . .... ... . ... . . . . .. . . .. ... .. . . . .. .. . . . . .. . ... ....................................................... . . . ......................................................................... ........................................ ..... . . ... . . . . . . .. . ...... .. . . .. ... . ... . ... . .... . ... . .. . . . . . . .... ... . .. . ..... . . . . . . . ...... . . ..... . ...... ......... . ...... ....... ...... . . ...... ........ ........ . ........ ................................. . . ..................... . . . . . . . . . .

1

x

Figure 5.2: Duffing equation with Γ = 0.28. (Period 2 solution.) eigenvalues, from which the Lyapunov exponents may be found. In general, da there are no stationary solutions and the equations dx dt = g and dt = M (x)a must be numerically solved simultaneously. See Wolf et al. [12] for a numerical technique for computing Lyapunov exponents.

Example Consider the Duffing equation: x ¨ + k x˙ − x + x3 = Γ cos ωt. This can be converted to an autonomous system as follows:     y x d    dx y = −ky + x − x3 + Γ cos z  . = (5.3) dt dt z ω Figures 5.1–5.3 show the different behavior of this system (x(t) versus t and x(t) versus y(t)) when k = 0.3, ω = 1.2, and Γ takes on the values 0.20, 0.28 and 0.50. For the numerical simulations shown, the initial conditions used were x0 = (1.3, 0, 0), and we began plotting the results when t = 50 to remove any initial transients. From deeper analysis, it can be shown that the system has a period 1 (2, 4, 5, 2, 1) solution when Γ = 0.20 (0.28, 0.29, 0.37, 0.65, 0.73). The solution is chaotic when Γ = 0.50. A different set of parameters is shown in figure 5.4. This figure has a plot of the three Lyapunov exponents of equation (5.3) when ω = 1.0, k = 0.5, and Γ is varied from 0.2 to 0.9. At low values of Γ, the system is periodic because the largest Lyapunov exponent is zero. The system follows c CD-ROM Handbook of Differential Equations Academic Press 1997

32

1 0

I.A . .. .. ..... ..... .. ... ........ ...... ... ... . ... .. .... .. ...... . ... ... ... .... ..... .... .. ... ... .. .... ... ... .... .... ... ... ... .... .. . . . ... .. ... .. ... ..... .... . . . . . .. . . . . .. .. .. .. .. .. ... ... ... ... ... ... ... .. .. ... ... ... ... ... .. .. ... .... ..... ... ... ... .. ... .. ... ..... .... ... .. .... . . ... .. .. ... ... .. ......... ... .. .. .. .. .. ... . .. .. ... .. .. ... .. .... ..... .....

75

t

100

Definitions and Concepts

1

... ......... ................... .... ... .................. ....................... ................... ... ... ............. .... ...................... .......... . . ..... .... ...... ..... ........................ ...... .......................... ......... ....... ... . ... .. ..... ..... .... ............ .. .... ..... . .... ...... ... .. .... ..... ... .... .... .. . ... ..... . . .. .. . . . . . . . . . ..... .. ... .. . .. .. ...... .. .. .. . . .. ... .. .. ........... .... . . . . ............................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................... .. .. . . .......... .. .. .. . ... ... .. .... .. .. ... .. . ... .. .. .... .. .. ... ....... .. .. ... .. ... .... ... ........................ ... .. ........ .... ............. ......... ........... .. .. . ................... ........... ...... ... ......... . . . . . . . . . ............... .............................................. ..................... ... ...... .... .. . .............................................. ..... ........ ...........

0

,1

1

Figure 5.3: Duffing equation with Γ = 0.50. (Chaotic solution.)

Figure 5.4: The three Lyapunov exponents for Duffing’s equation with ω = 1.0 and k = 0.5 when Γ is varied from 0.2 to 0.9. (From De SouzaMachado, S., Rollins, R. W., Jacobs, D. T., & Hartman, J. L. Studying chaotic systems using microcomputer simulations and Lyapunov exponents. Amer. J. Physics 58, 4, April 1990, 321–329.)

a period doubling route to chaos at Γ ≈ 0.36, when the largest Lyapunov exponent becomes greater than zero. The system remains chaotic until the driving force gets very large (Γ > 0.84) except for windows of periodicity, which occur throughout the chaotic regime.

Notes 1. There are at least three scenarios in which the regular behavior of a system becomes chaotic. A standard route is via a series of perioddoubling bifurcations. Two other routes to chaos that are fairly well understood are via intermittent behavior and through quasiperiodic solutions. 2. Many equations have been shown to be chaotic: • Hale and Sternberg [4] have shown that the differential delay x(t−τ ) = ax(t) + b 1+x equation dx(t) n (t−τ ) is chaotic for certain padt rameter regimes. c CD-ROM Handbook of Differential Equations Academic Press 1997

x

5.

Chaos in Dynamical Systems ................... .. ... ... ... .. .. ............ ....... .. ......... ................. ... .... ...

C2

............ ................................... ... ................................. ............ . . . . . . .... ........

G

L

33

R C1

GN

i g

. . . .... .... ... .... .... ..... .. ............... .. ..... ..... .... .... ..... ....

GN

............ ........ .................... ... .. ...... .. .. ..................... .............. .. .....................

..... ... ... ... .. .. . . . . . . . . . . . . . . ... . . ..... . .. .... . . .. ..... . . . . .... .. . .. .. .... . . . . . .. ... . .. .... . ... . .. ..... .... . .. . . ....... ........................................................................................................................................ ... .. ...... ... ...... . .. . .. . ...... .. .. . . . . . . ..... .. . ... . . . . . ... . .. . .... .... . ... ... . . . ...... . . . . . . . . . . . .

,1 ,g

1

v

Figure 5.5: The canonical piecewise-linear circuit and the voltage-current characteristic of the nonlinear resistor GN . • The equations defining the Lorenz attractor are x˙ = 10y − 10x, y˙ = −y − xz + 28x, 8 z˙ = xy − z. 3

(5.4)

• The R¨ ossler equations are x˙ = −(y + z), y˙ = x + ay, z˙ = b + xz − cz.

(5.5)

When a = 0.343, b = 1.82, and c = 9.75, this generates the “R¨ossler funnel.” When a = 0.2, b = 0.2, and c = 5.7, this generates “the simple R¨ ossler attractor.” 3. For an autonomous electronic circuit to exhibit chaos, it must contain at least three energy storage devices. (Otherwise, the Poincar´e– Bendixson theorem states that the limiting set will be a point or a limit cycle, not a strange attractor.) A simple circuit with three energy storage devices that produces chaos is in Matsumoto [7]. The circuit given in Chua and Lin [2] (see figure 5.5) is almost as simple as that given by Matsumoto and can simulate (by choosing different values for the nonlinear resistor) different chaotic phenomena in a large three-dimensional state space. This circuit contains only six two-terminal elements: Five of them are linear resistors, capacitors, and inductors; and one element (GN ) is a three-segment, piecewiselinear resistor. 4. Different types of dynamical systems can have greater or lesser degrees of randomness. A simple classification of the amount of randomness in dynamical systems is as follows: • Ergodic systems: this is the “weakest” level of randomness, in which phase averages equal time averages. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

• Mixing systems: here, no time averaging is required to reach “equilibrium.” • K-systems: systems with positive Kolmogorov entropy. This means that a connected neighborhood of trajectories must exhibit a positive average rate of exponential divergence. • C-systems: every trajectory has a positive Lyapunov exponent. • Bernoulli systems: these systems are as random as a fair coin toss. See Tabor [11] for details. 5. A technical definition of the Lyaponuv exponents is as follows: When A(t) is a bounded coefficient matrix, consider the n-dimensional linear system y0 = A(t)y(t). Consider n linearly independent solutions of this in the form yi = Y (t)pi , where Y (t) is a fundamental solution matrix with Y (0) orthogonal, and the {pi } form an orthonormal basis. The characteristic numbers are defined as λi = lim sup t→∞

6.

7.

8.

9.

1 log (||Y (t)pi ||) . t

When the sum of the characteristic numbers is minimized, the orthogonal basis {pi } is called normal and the {λi } are the Lyapunov exponents. There are many software packages for numerically computing Lyapunov exponents. See, for example, Parker and Chua [8] and Rollins [9]. In this section we have focused on chaos appearing in coupled, firstorder, ordinary differential equations. Chaos can also appear in partial differential equations and stochastic equations. The papers by Ablowitz and Herbst [1], Lorenz [6], and and Yamaguti and Ushiki [13] describe and illustrate how numerical discretizations of a differential equation can lead to discrete equations exhibiting chaos. By long-term integration of the equations governing the solar system on speical purpose computers, researchers have found that Pluto’s orbit is chaotic, the motion of the Jovian planet subsystem is chaotic, and the motion of comet Halley is chaotic. See, for example, Sussman and Wisdom [10].

References [1] Ablowitz, M. J., and Herbst, B. M. On homoclinic structure and numerically induced chaos for the nonlinear Schrodinger equation. SIAM J. Appl. Math. 50, 2 (April 1990), 339–351. [2] Chua, L. O., and Lin, G.-N. Canonical realization of Chua’s circuit family. IEEE Trans. Circ. & Syst. 37, 7 (July 1990), 885–902. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Chaos in Dynamical Systems

35

[3] Geist, K., Parlitz, U., and Lauterborn, W. Comparison of different methods for computing Lyapunov exponents. Progr. Theoret. Phys 83, 55 (1990), 875–893. [4] Hale, J. K., and Sternberg, N. Onset of chaos in differential delay equations. J. Comput. Physics 77 (1988), 221–317. [5] Hassard, B., Zhang, J., Hastings, S. P., and Troy, W. C. A computer proof that the Lorenz equations have “chaotic” solutions. Appl. Math. Lett. 7, 1 (1994), 79–83. [6] Lorenz, E. N. Computational chaos—A prelude to computational instability. Physica D 35 (1989), 299–317. [7] Matsumoto, T. Chaos in electronic circuits. Proc. IEEE, 8 (August 1987). [8] Parker, T. S., and Chua, L. O. Efficient solution of the variational equation for piecewise-linear differential equations. Circuit Theory and Appl. 14, 4 (1986), 305–314. [9] Rollins, R. W. Chaotic Dynamics Workbench. Tech. rep., Physics Academic Software, AIP, New York, 1990. [10] Sussman, G. J., and Wisdom, J. Chaotic evolution of the solar system. Science 257, 5066 (1992), 56–62. [11] Tabor, M. Chaos and Integrability in Nonlinear Dynamics. John Wiley & Sons, New York, 1989. [12] Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A. Determining Lyapunov exponents from a time series. Physica D 16 (1985), 285–317. [13] Yamaguti, M., and Ushiki, S. Chaos in numerical analysis of ordinary differential equations. Physica D 3 (1981), 618–626.

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6.

Definitions and Concepts

Classification of Partial Differential Equations

Applicable to

Partial differential equations.

Yields Knowledge of the type of equation under consideration.

Procedure Most partial differential equations are of three basic types: elliptic, hyperbolic, and parabolic. Elliptic equations are often called potential equations. They result from potential problems, where the potential might be temperature, voltage, or a similar quantity. Elliptic equations are also the steady solutions of diffusion equations, and they require boundary values in order to determine the solution. Hyperbolic equations are sometimes called wave equations, because they often describe the propagation of waves. They require initial conditions (where the waves start from) as well as boundary conditions (to describe how the wave and the boundary interact; for instance, the wave might be scattered or absorbed). These equations can be solved, in principle, by the method of characteristics (see page 432). Parabolic equations are often called diffusion equations because they describe the diffusion and convection of some substance (such as heat). The dependent variable usually represents the density of the substance. These equations require initial conditions (what the initial concentration of the substance is) as well as boundary conditions (to specify, for instance, whether the substance can cross the boundary or not). The above classification is most useful for second order partial differential equations. For second order equations, only characteristic curves need to be considered. For equations of higher degree, characteristic surfaces must be considered, see Whitham [8, pages 139–141] or Zauderer [10, pages 78–85 and 91–97] for more details. After two special cases, we specialize the rest of this section to second order partial differential equations.

Special Case 1 The most general second order linear partial differential equation with constant coefficients n n X X ∂2u ∂u aij + bi + cu = d, ∂x ∂x ∂x i j i i,j=1 i=1 may be placed in the form uξ1 ξ1 + · · · + uξn ξn + λu = 0, c CD-ROM Handbook of Differential Equations Academic Press 1997

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Classification of Partial Differential Equations

37

if the equation is elliptic or may be placed in the form uξ1 ξ1 − uξ2 ξ2 − · · · − uξn ξn + λu = 0, if the equation is hyperbolic, for some value of λ. See Garabedian [3, pages 70–76] for details.

Special Case 2 The (real valued) second order partial differential equation in n dimensions   n X ∂2u ∂u ∂u = 0, (6.1) aij (x) + f x, u, ,..., ∂xi ∂xj ∂x1 ∂xn i,j=1 for u(x) = u(x1 , . . . , xn ), where aij = aji , may be classified at the point x0 as follows. Let A be the matrix (aij (x0 )). By means of a linear transformation, the quadratic form gT Ag may be reduced to the form λ1 g12 + λ2 g22 + · · · + λn gn2 . The values of {λi }, which are the eigenvalues of A, determine the nature of the partial differential equation (6.1). Because A has been assumed to be symmetric, all of the eigenvalues will be real. The classification at the point x0 is then given by 1. If all of the {λi } are of the same sign, then equation (6.1) is elliptic at x0 . 2. If any of the {λi } are zero, then equation (6.1) is parabolic at x0 . 3. If none of the {λi } are zero and they are not all of the same sign, then equation (6.1) is hyperbolic at x0 . 4. If none of the {λi } are zero and there are at least two that are positive and at least two that are negative, then equation (6.1) is ultrahyperbolic at x0 . If an equation is parabolic along a smooth curve in a domain D, and the equation is hyperbolic on one side of the curve and elliptic on the other side of the curve, then the equation is of mixed type. The smooth curve is called the curve of parabolic degeneracy.

Special Case 3 We further specialize here and restrict ourselves to second order equations in two independent variables. Consider partial differential equations of second order in two independent variables, of the form   ∂2u ∂2u ∂u ∂u ∂2u + C(x, y) 2 = Ψ u, , , x, y , A(x, y) 2 + B(x, y) ∂x ∂x∂y ∂y ∂x ∂y (6.2) where Ψ need not be a linear function. c CD-ROM Handbook of Differential Equations Academic Press 1997

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  2   B − 4AC > 0 hyperbolic If B 2 − 4AC = 0 at some point (x, y), then equation (6.2) is  parabolic  elliptic B 2 − 4AC < 0 at that point. If an equation is of the same type at all points in the domain, then the equation is simply said to be of that type. Equation 6.2 can be transformed into a canonical form for each of the three types mentioned above. The procedures are as follows.

Hyperbolic Equations For hyperbolic equations we look for a new set of independent variables ζ = ζ(x, y) and η = η(x, y) for which equation (6.2) may be written in the standard form uζη = φ (u, uη , uζ , η, ζ) .

(6.3)

Utilizing this change of variables, we can calculate ux = uη ηx + uζ ζx , uy = uη ηy + uζ ζy , uxx = uηη ηx ηx + 2uηζ ηx ζx + uζζ ζx ζx + uη ηxx + uζ ζxx , uxy = uηη ηx ηy + 2uηζ (ηx ζy + ηy ζx ) + uζζ ζx ζy + uη ηxy + uζ ζxy , uyy = uηη ηy ηy + 2uηζ ηy ζy + uζζ ζy ζy + uη ηyy + uζ ζyy , to find that equation (6.2) transforms into Auζζ + Buζη + Cuηη = Φ (u, uη , uζ , η, ζ) ,

(6.4)

where A = Aζx2 + Bζx ζy + Cζy2 , B = Aζx ηx + B(ζx ηy + ζy ηx ) + 2Cζy ηy , C = Aηx2 + Bηx ηy + Cηy2 . Setting A = C = 0, we can find the following partial differential equations for ζ and η √ ζx −B + B 2 − 4AC , = ζy 2A (6.5.a-b) √ −B − B 2 − 4AC ηx . = ηy 2A These equations may be readily solved (in principle) by the method of characteristics. For example, to solve equation (6.5.a) we only need to solve √ dy −B + B 2 − 4AC − = dx 2A c CD-ROM Handbook of Differential Equations Academic Press 1997

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Classification of Partial Differential Equations

39

for Q(x, y) = R, where R is an arbitrary constant. Then ζ will be given by ζ = Q(x, y). After ζ and η are determined, then the original equation must be transformed into the new coordinates (see page 168). The resulting equation will then be in standard form. Note that another standard form for hyperbolic equations (in two independent variables) is obtained from equation (6.3) by the change of variables α = η − ζ,

β = η + ζ.

(6.6)

This results in the equation   1 1 uαα − uββ = φ u, uα − uβ , uα + uβ , (β + α) , (β − α) . 2 2

Example 1 Suppose we have the equation y 2 uxx − x2 uyy = 0.

(6.7)

We recognize this equation to be hyperbolic away from the lines x = 0 and y = 0. To find the new variables ζ and η, we must solve the differential equations in (6.5). For this equation, we have {A = y 2 , B = 0, C = −x2 }. Therefore (6.5) becomes x ζx =− , ζy y

ηx x = , ηy y

with the solutions ζ = y 2 −x2 , η = y 2 +x2 . In these new variables, equation (6.7) becomes uζη =

ζ η uη − uζ . 2(ζ 2 − η 2 ) 2(ζ 2 − η 2 )

(6.8)

If the change of independent variable in (6.6) is made, then (6.8) becomes uαα − uββ =

1 1 uβ − uα . 2β 2α

Parabolic Equations For parabolic equations, we look for a new set of variables ζ = ζ(x, y) and η = η(x, y) in which equation (6.2) can be written in one of the standard forms uζζ = φ (u, uη , uζ , η, ζ) ,

(6.9.a)

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Definitions and Concepts

or uηη = φ (u, uη , uζ , η, ζ) .

(6.9.b)

Utilizing equation (6.4), we see that we need to determine ζ and η in such a way that B = 0 = C,

corresponding to (6.9.a),

(6.10.a)

B = 0 = A,

corresponding to (6.9.b),

(6.10.b)

or

If A 6= 0, then equation (6.10.a) corresponds to the single equation B ζx =− , ζy 2A

(6.11.a)

while, if C 6= 0, then equation (6.10.b) corresponds to the equation ζx B . =− ζy 2C

(6.11.b)

In either case, we have only to solve a single equation to determine ζ. The variable η can then be chosen to be anything linearly independent of ζ. As before, once ζ and η are determined, then the equation needs to be written in terms of these new variables

Example 2 Suppose we have the equation y 2 uxx − 2xyuxy + x2 uyy + uy = 0.

(6.12)

Since {A = y 2 , B = −2xy, C = x2 }, we find that B 2 − 4AC = 0 and so this equation is parabolic. In this case we choose to make B = C = 0. From equation (6.11.a) we must solve ζζxy = xy , which has the solution ζ = y 2 +x2 . We choose η = x. Using these values of η and ζ, we find that (6.12) becomes 2(ζ + η) 1 uηη = uζ + uη . 2 ζ −η ζ − η2

Elliptic Equations For elliptic equations we look for a new set of variables α = α(x, y) and β = β(x, y) in which equation (6.2) can be written in the standard form uαα + uββ = φ (u, uα , uβ , α, β) . c CD-ROM Handbook of Differential Equations Academic Press 1997

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The easiest way in which to find α and β is to determine variables ζ = ζ(x, y) and η = η(x, y) that satisfy √ (6.5) and then form α = (η + ζ)/2, β = (η − ζ)/2i (where, as usual, i = −1). Note that in this case, the differential equations in (6.5) are complex. However, since ζ and η are conjugate complex functions, the quantities α and β will be real.

Example 3 Suppose we have the equation y 2 uxx + x2 uyy = 0. We recognize this equation to be elliptic away from the lines x = 0 and y = 0. To find the new variables ζ and η, we must solve the differential equations in (6.5). For this equation, we have {A = y 2 , B = 0, C = x2 }. Therefore (6.5) becomes ix ζx =− , ζy y

ηx ix = , ηy y

with the solutions ζ = y 2 − ix2 , η = y 2 + ix2 . Forming α and β results in α=

η+ζ = y2, 2

β=

η−ζ = x2 . 2i

In these new variables, equation (6.7) becomes uαα + uββ = −

1 1 uα − uβ . 2α 2β

Notes 1. Equations of mixed type are discussed in Haack and Wendland [4] and Smirnoff [6]. 2. Given a partial differential equation in the form of equation (6.1), the characteristic surfaces are defined by the characteristic equation    n X ∂u ∂u = 0. aij (x) ∂xi ∂xj i,j=1 The solutions to this equation are the only surfaces across which u(x) may have discontinuities in its second derivatives. 3. The Notes section of the characteristics method (see page 432) describes how to determine when a system of partial differential equations is hyperbolic. 4. See also Farlow [2, pages 174–182 and 331–339], Moon and Spencer [5, pages 137–146], Stakgold [7, pages 467–482], and Young [9, pages 60–70].

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References [1] Bitsadze, A. V. Equations of the Mixed Type. The MacMillan Company, New York, 1964. [2] Farlow, S. J. Partial Differential Equations for Scientists and Engineers. John Wiley & Sons, New York, 1982. [3] Garabedian, P. R. Partial Differential Equations. John Wiley & Sons, New York, 1964. [4] Haack, E. R., and Wendland, W. N. Lectures on Partial and Pfaffian Differential Equations. Pergamon Press, New York, 1972. Translated by E. R. Dawson and W. N. Everitt. [5] Moon, P., and Spencer, D. E. Partial Differential Equations. D. C. Heath and Co., Lexington, MA, 1969. [6] Smirnoff, M. M. Equations of Mixed Type. Amer. Math. Soc., Providence, RI, 1978. [7] Stakgold, I. Green’s Functions and Boundary Value Problems. John Wiley & Sons, New York, 1979. [8] Whitham, G. B. Linear and Nonlinear Waves. Interscience Publishers, Inc., New York, 1974. [9] Young, E. C. Partial Differential Equations. Allyn and Bacon, Inc., Boston, MA, 1972. [10] Zauderer, E. Partial Differential Equations of Applied Mathematics. John Wiley & Sons, New York, 1983.

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7.

Compatible Systems

7.

43

Compatible Systems

Applicable to

Systems of differential equations.

Yields Knowledge of whether the equations are consistent.

Procedure 1 The two equations f (x, y, z, p, q) = 0 and g(x, y, z, p, q) = 0 for z = z(x, y) (where, as usual, p = zx and q = zy ) are said to be compatible if every solution of the first equation is also a solution of the second equation, and conversely. These two equations will be compatible if {f, g} = 0, where {f, g} := and where

∂(u,v) ∂(a,b)

∂(f, g) ∂(f, g) ∂(f, g) ∂(f, g) +p + +q , ∂(x, p) ∂(z, p) ∂(y, q) ∂(z, q)

= | uuab

va vb

| = ua vb − va ub is the usual Jacobian.

Procedure 2 The conditions for consistency of a system of simultaneous partial differential equations of the first order, if the number of equations is an exact multiple of the number of dependent variables involved, is given in Forsyth [3, Part IV, pages 411–419]. To write the consistency conditions, let the unknown dependent variables be {zi | i = 1, . . . , m}, let the independent variables be {xj | j = 1, . . . , n}, and define pij = ∂zi /∂xj . We presume the system has rm equations (with r ≤ n) and that these equations can be solved with respect to the pij . That is pij =

∂zi = fij ({xl } , {zk } , {pλµ }) , ∂xj

for i = h1, mi, j = h1, ni, l = h1, ni, λ = h1, mi, µ = hr + 1, ni. (Here we have introduced the notation ha, bi to be the sequence of numbers a, a + 1, a + 2, . . . , b.) Then, for consistency, the following conditions must be satisfied  m  ∂fia X ∂fij ∂fij ∂fia fλa − + − fλj ∂xa ∂xj ∂zλ ∂zλ λ=1   m n X X ∂fia ∂fsj ∂fij ∂fsa − + ∂psµ ∂xµ ∂psµ ∂xµ s=1 µ=r+1 (7.1)    n m m X X X ∂fia ∂fsj ∂fij ∂fsa pλµ = 0, + − ∂psµ ∂zλ ∂psµ ∂zλ s=1 µ=r+1 λ=1

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Definitions and Concepts

where i = h1, mi, a = hj + 1, ri, j = h1, r − 1i, and  m  X ∂fia ∂fsj ∂fij ∂fsa ∂fia ∂fsj ∂fij ∂fsa = 0, − + − ∂psµ ∂pkτ ∂psµ ∂pkτ ∂psτ ∂pkµ ∂psτ ∂pkµ s=1

(7.2)

where i, k = h1, mi, a = hj + 1, ri, µ, τ = hr + 1, ni, j = h1, r − 1i.

Special Case 1 In the special case of m = 1, we have one dependent variable (which we call z) and r equations. Let pj = ∂z/∂xj = fj (z, x1 , . . . , xn , pr+1 , . . . , pn ). In this case, equation (7.2) is automatically satisfied while equation (7.1) becomes   n X dfa ∂fa dfj dfj ∂fj dfa =0 − + − dxa dxj µ=r+1 ∂pµ dxµ ∂pµ dxµ for a = h1, j − 1i, j = h1, ri, where we have defined

d dxs

=

∂ ∂xs

∂ + ps ∂z .

Special Case 2 In the special case of r = n, the system of mn equations becomes pij = fij (z1 , . . . , zm , x1 , . . . , xn ) and the consistency conditions become  m  ∂fia X ∂fij ∂fia ∂fij fλa =0 − + − fλj ∂xa ∂xj ∂zλ ∂zλ λ=1

for i = h1, mi, a = h1, j − 1i and j = h1, ni. These are known as Mayer’s system of completely integrable equations.

Special Case 3 Consider the special case of r = 1, with {F1 = 0, F2 = 0, . . . , Fm = 0}, where each Fj = pj − fj (z, x1 , . . . , xn , pr+1 , . . . , pn ) is analytical in each of its arguments. A necessary and sufficient condition for the set of equations to be consistent is that [Fi , Fj ] = 0, for all combinations of i and j. Here, [ , ] represents the usual Poisson bracket.

Example Suppose we have the two following nonlinear partial differential equations for z(x, y): xzx = yzy ,

z(xzx + yzy ) = 2xy.

(7.3)

From (7.3) we identify f (x, y, z, p, q) = xp − yq,

g(x, y, z, p, q) = z(xp + yq) − 2xy. (7.4)

Using (7.4) we can easily calculate ∂(f, g) = 2xy, ∂(x, p)

∂(f, g) = −x2 p, ∂(z, p)

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∂(f, g) ∂(f, g) = −2xy, = xyp. ∂(y, q) ∂(z, q) Therefore, computing {f, g}, we find it to be zero. Hence, the two equations in equation (7.3) have identical solution sets. Because the equations in (7.3) are compatible, we can combine them without changing the solution sets. Solving the equations in (7.3) simultaneously for p and q to obtain {zx = p = y/z, zy = q = x/z}. These last two equations can be easily solved we obtain z 2 = C + 2xy, where C is an arbitrary constant.

Notes 1. Jacobi’s method (see page 464) takes a given partial differential equation and creates a compatible equation and then uses elimination between these two equations. 2. If it is known that a linear homogeneous ordinary differential equation of order n has solutions in common with a linear homogeneous ordinary differential equation of order m (with m < n), then it is possible to determine a differential equation of lower degree that has, as its solutions, these common solutions. If the linear homogeneous ordinary differential equations L1 [u] = 0 and L2 [u] = 0 are defined by L1 := p0 Dn + p1 Dn−1 + · · · + pn−1 D + pn , L2 := q0 Dm + q1 Dm−1 + · · · + qm−1 D + qm , where D represents d/dx and each of the functions {pi , qi } depends on x, define the ordinary differential equation R1 [u] = 0 by R1 := r0 Dn−m + r1 Dn−m−1 + · · · + rn−m−1 D + rn−m , where the {ri } are defined by p0 = r0 q0 ,

   n−m 0 q0 + q1 , 1    n−m−1 0 p2 = r2 q0 + r1 q0 + q1 1      n − m 00 n−m 0 q0 + q1 + q2 , + r0 2 1 .. .    1 0 pn−m = rn−m q0 + rn−m−1 q + q1 1 0        2 00 2 0 2 q + q + q2 + . . . , + rn−m−2 2 0 1 1 0 p1 = r1 q0 + r0

= rn−m q0 + rn−m−1 [q00 + q1 ] + rn−m−2 [q000 + 2q10 + q2 ] + . . . . c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

Then the order of the operator L3 := L1 − R1 L2 will be depressed as much as is possible (the order of L3 will not exceed m − 1). Note that only a finite number of rational operations and differentiations are required to determine the {ri }. From the definition of L3 , we see that all solutions common to both L1 [u] = 0 and to L2 [u] = 0 will also be solutions to L3 [u] = 0. If L3 is identically zero, then we have found a factorization of L1 (see page 294). See Ince [4, pages 126–128] or Valiron [6, pages 320–322] for details. 3. Differential resultants can also be used to derive consistency conditions. See Berkovich and Tsirulik [2] for details. 4. Wolf [7] describes an algorithm that determines if an overdetermined system of two equations for one function has any solution. An implementation in FORMAC is mentioned. 5. See also Ames [1, pages 54–65] and Sneddon [5, pages 67–68].

References [1] Ames, W. F., Ed. Nonlinear Partial Differential Equations, vol. 1. Academic Press, New York, 1967. [2] Berkovich, L. M., and Tsirulik, V. G. Differential resultants and some of their applications. Differentsial’nye Uravneniya 22, 5 (May 1986), 750–757. [3] Forsyth, A. R. Theory of Differential Equations. Dover Publications, Inc., New York, 1959. [4] Ince, E. L. Ordinary Differential Equations. Dover Publications, Inc., New York, 1964. [5] Sneddon, I. N. Elements of Partial Differential Equations. McGraw–Hill Book Company, New York, 1957. [6] Valiron, G. The Geometric Theory of Ordinary Differential Equations and Algebraic Functions. Math Sci Press, Brookline, MA, 1950. [7] Wolf, T. An analytic algorithm for decoupling and integrating systems of nonlinear partial differential equations. J. Comput. Physics 60 (1985), 437– 446.

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8.

Conservation Laws

8.

47

Conservation Laws

Applicable to

Partial differential equations.

Yields Quantities that remain invariant during the evolution of the partial differential equation.

Procedure Given an evolution equation, which is a partial differential equation of the form ut = F (u, ux , uxx , . . . ),

(8.1)

a conservation law is a partial differential equation of the form   ∂  ∂  T u(x, t) + X u(x, t) = 0, (8.2) ∂t ∂x which is satisfied by all solutions of equation (8.1). We define T (·) to be the conserved density and X(·) to be the flux. An alternative statement of equation (8.2) is that Z   T u(x, t) dx (8.3) is independent of t, for solutions of (8.1) such that the integral converges. More generally, a partial differential equation of order m in the n independent variables x = (x1 , x2 , . . . , xn ) and a single dependent variable u is in conservation form if it can be written as n X ∂ Fi (x, u, ∂u, ∂ 2 u, . . . , ∂ m−1 u) = 0. ∂x i i=1

(8.4)

Here ∂ j u represents all jth order partial derivatives of u with respect to x.

Example 1 The Korteweg–de Vries equation ut = uxxx + uux

(8.5)

has an infinite set of conservation laws. The first few, in order of increasing rank, have the conserved densities T = u, T = u2 , T = u3 − 3u2x , T = 5u4 − 60uu2x − 36uxuxxx , .. . c CD-ROM Handbook of Differential Equations Academic Press 1997

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To demonstrate, for instance, that T = u2 is a conserved density, we compute ∂(u2 ) ∂T = = 2uut = 2uuxxx + 2u2 ux , ∂t ∂t where we have used the defining equation in (8.5) to replace the ut term. Now we must determine a flux X such that equation (8.2) is satisfied. In this case, we find X = u2x − 2uuxx − 23 u3 .

Example 2 The Schr¨ odinger equation −

∂2u ∂u + V (x)u = i ∂x2 ∂t

can be expressed in the form of equation (8.2) with T = iν(x)u, ∂u − ν 0 (x)u, X = ν(x) ∂x where ν(x) is defined by ν 00 (x) = V (x)ν(x).

Notes 1. Conservation laws allow estimates of the accuracy of a numerical solution scheme (because the quantity in (8.3) must be invariant in time). 2. Not all partial differential equations have an infinite number of conservation laws; there may be none or a finite number. 3. A conservation law for an evolution equation is called trivial if T is, itself, the x derivative of some expression. If equation (8.1) has an infinite sequence of nontrivial conservation laws, then the equation is formally integrable. Infinite sequences of nontrivial conservation laws are given by Cavalcante and Tenenblat [2] for the following equations: Burgers, KdV, mKdV, sine–Gordon, sinh–Gordon. 4. If a given partial differential equation is not written in conservation form, there are a number of ways of attempting to put it in a conserved form. Bluman er al. [1] have a short list of techniques. 5. If equation (8.4) is satisfied, then there exists an (n − 1)-exterior differential form F such that equation (8.4) can be written dF = 0. This implies that there is an (n − 2)-form φ such that F = dφ. This, in turn, means that there exists an antisymmetric tensor of rank n, ψ, such that Fi (x, u, ∂u, ∂ 2 u, . . . , ∂ m−1 u) =

X i<j≤n

(−1)j

X ∂ψij ∂ψji + (−1)i−1 , ∂xj ∂xj 1≤j
for i = 1, 2, . . . , n. c CD-ROM Handbook of Differential Equations Academic Press 1997

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6. A computer program in REDUCE for determining conservation laws is given in Ito and Kako [6]. In Gerdt et al. [4] is the description of a computer program in FORMAC that determines conservation laws, determines Lie–B¨ acklund symmetries, and also attempts to determine when an evolution equation is formally integrable. 7. Torriani [10] shows how the terms appearing in the expression of the densities and the fluxes for the Korteweg-de Vries equation may be found by combinatorial methods. 8. El-Sherbiny [3] proves that unless a1 /a2 is a multiple root of order three of the algebraic equation a6 λ3 − a5 λ2 + a4 λ − a3 = 0, then the class of nonlinear evolution equations ut + ux + a1 uux + a2 uut + a3 uxxx + a4 uxxt + a5 uxtt + a6 uttt = 0 with the {ai } real numbers has a finite number of conservation laws; otherwise, the class has an infinite number of conservation laws. 9. See also Olver [9, Chapter 4, pages 246–291].

References [1] Bluman, G. W., Reid, G. J., and Kumei, S. New classes of symmetries for partial differential equations. J. Math. Physics 29, 4 (April 1988), 806–811. [2] Cavalcante, J. A., and Tenenblat, K. Conservation laws for nonlinear evolution equations. J. Math. Physics 29, 4 (April 1988), 1044–1049. [3] El-Sherbiny, H. M. Conservation laws of a class of differential equations II. Appl. Math. Lett. 6, 4 (1993), 99–103. [4] Gerdt, V. P., Shvachka, A. B., and Zharkov, A. Y. Computer algebra applications for classification of integrable non-linear evolution equations. J. Symbolic Comp. 1 (1985), 101–107. [5] Ibragimov, N. H. Group theoretical nature of conservation theorems. Letters Math. Physics 1 (1977), 423–428. [6] Ito, M., and Kako, F. A REDUCE program for finding conserved densities of partial differential equations with uniform rank. Comput. Physics Comm. 38 (1985), 415–419. [7] LeVeque, R. J. Numerical Methods for Conservation Laws. Birkhauser, Basel, Switzerland, 1992. [8] Littinsky, E. Polynomial integrals of evolution equations. Comm. Math. Physics 121 (1989), 669–682. [9] Olver, P. J. Applications of Lie Groups to Differential Equations. No. 107 in Graduate Texts in Mathematics. Springer–Verlag, New York, 1986. [10] Torriani, H. H. Conservation laws for the Korteweg-de Vries equation and the theory of partitions. Physics Letters 113A, 7 (6 January 1986), 345–348. [11] Vinokur, M. An analysis of finite-difference and finite-volume formulations of conservations laws. J. Comput. Physics 81 (1989), 1–52.

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9.

Definitions and Concepts

Differential Resultants

Applicable to

Two polynomial ordinary differential equations.

Yields One ordinary differential equation in one independent variable.

Idea Given two polynomial equations (in, say, x and y), the classical method of resultants is as follows: The equations can always be written as the system of linear equations Aw = 0, where A = A(y) and w = w(x) 6= 0. Because this system must have det A = 0, a polynomial equation only in y may be determined. The technique for polynomial differential equations is very similar.

Procedure Resultants have classically been used to eliminate one variable between two polynomial equations. For example, suppose we have the two equations x3 − 3y 2 x2 + x + 5y 2 = 0,

(9.1)

x3 + 5y 2 x2 − x + 3y 2 = 0.

These equations may be multiplied by powers of x to obtain the system of equations: x5

x5

−

+

3y 2 x4 x4

+ −

x4 5y 2 x4

+ −

x3 3y x x3 x3 5y 2 x3 x3 2 3

+ + − + − +

5y 2 x2 x2 2 2 3y x 5y 2 x2 x2 3y 2 x2

+ 5y 2 x + x + − x + + 3y 2 x

= = 5y 2 = 3y 2 = = =

0, 0, 0, 0, 0, 0.

This system can be written in matrix form as  1 −3y 2 0 1  0 0  0 0  0 1 1 5y 2

1 −3y 2 1 1 5y 2 −1

5y 2 1 −3y 2 5y 2 −1 3y 2

0 5y 2 1 −1 3y 2 0

  5   0 x 0 x4  0 0    3       5y 2   x  0 2  2  =   . 3y  x  0 0   x  0 0 1 0

(9.2)

This last equation is a 6 × 6 system of the form Aw = 0. Because w 6= 0 (because, at least, the last component of w is non-zero), the determinant of A must vanish. Taking the determinant of the matrix in equation (9.2), we find that y must satisfy the equation 32y 2 (289y 8 + 16y 4 + 1) = 0.

(9.3)

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Differential Resultants

51

All the different values of y, from the solutions of (9.1), must satisfy (9.3). Differential resultants are the analogue of resultants applied to differential systems. There are two steps analogous to multiplying the original equations by powers of x. They are • differentiating one of the equations, • multiplying one of the equations by some term that may involve the independent and/or the dependent variables. Although there are algorithms published on how to proceed in any given case, as in Mishina and Proskuryakov [3], they are generally written in the language of abstract algebra.

Example Suppose we have the following two coupled differential equations for {y(x), z(x)} A:

3yz + z − yx = 0,

B:

− zx + z 2 + y 2 + y = 0.

We seek a single differential equation involving only z(x). Note that we could solve equation (A) for y(x) (by integrating factors) and then substitute this result in equation (B), but this creates an algebraic mess. This, in turn, makes it difficult to obtain a single simple equation for z(x). If we form the equations {A, B, yA, yB, yx B, ∂x B, y∂x A}, then we obtain the system 

0 1  3z  1  0  0 0

0 0 0 1 0 0 0

−1 0 0 0 z 2 − zx 1 0

0 0 −1 0 1 2 1

0 0 0 0 1 0 2

3z 1 z z 2 − zx 0 0 2zzx − zxx

z z 2 − zx 0 0 0 2zzx − zxx 0

   y2 0   y 3  0       yx  0       yyx  = 0 .  2     y yx  0       y  0 0 1 

Taking the determinant of the matrix above, we conclude that z(x) is a solution of the single ordinary differential equation 2 zxx + (−16zx + 12z 2 − 3)zzxx + 64z 2 zx2 + (23 − 96z 2)z 2 zx

+ (36z 4 − 17z 2 + 2)z 2 = 0.

Notes 1. This technique applies directly to systems of partial differential equations and to higher order equations. c CD-ROM Handbook of Differential Equations Academic Press 1997

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2. There are specific technical requirements for when the classical method of resultants (when applied to polynomials) will work. There are similar requirements for when differential resultants will work. See Mishina and Proskuryakov [3] for details. 3. Rubel [5] proves the following theorem, which indicates that elimination is not always possible, at least for algebraic differential equations (ADEs, see page 720): There exists a system of two ADEs, in the two dependent variables u and v which possesses a real-valued C n,m solution u, v on a certain open interval I, but which has no solution u, v on I for which v satisfies an ADE that does not involve u or any derivative of u. 4. By taking equations pairwise a system of, say, 10 equations in 10 different independent variables could, if fortunate, be reduced to a single equation in a single independent variable. 5. The two differential equations considered do not both have to be polynomial for this reduction scheme to work. The two equations have only to be polynomials in one of the dependent variables (the one that will be removed). 6. Any linear second order ordinary differential equation system can be interpreted as the resultant of the elimination of a dependent variable from a pair of conjugate first order Hamilton’s equations. See Tolstoy [7] for details.

References [1] Berkovich, L. M., and Tsirulik, V. G. Differential resultants and some of their applications. Differentsial’nye Uravneniya 22, 5 (May 1986), 750–757. [2] Bocher, M. Introduction to Higher Algebra. The MacMillan Company, New York, 1907. [3] Mishina, A. P., and Proskuryakov, I. V. Higher Algebra. Pergamon Press, New York, 1965. [4] Ritt, J. F. Differential Algebra. Dover Publications, Inc., New York, 1966. [5] Rubel, L. A. An elimination theory for systems of algebraic differential equations. Houston J. Math. 8, 2 (1982), 289–295. [6] Seidenberg, A. An elimination theory for differential algebra. University of California Press 3, 2 (1956), 31–66. [7] Tolstoy, I. Remarks on the linearization of differential equations. J. Inst. Maths. Applics 20 (1977), 53–60. [8] Van Der Waerden, B. L. Modern Algebra. Frederick Ungar Publishing, New York, 1940.

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Existence and Uniqueness Theorems

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53

Existence and Uniqueness Theorems

Applicable to

Differential equations of all types.

Yields Knowledge of whether a solution exists and, if so, if the solution is unique.

Idea There are theorems available for many cases of interest.

Procedure Corresponding to the difficulty of the subjects involved, there are more theorems applicable to: ordinary differential equations than partial differential equations, linear equations than nonlinear equations, and initial value problems than boundary value problems. In the following we indicate some of the simple theorems that are frequently useful. The last theorem is applicable to partial differential equations; the rest are applicable to ordinary differential equations. The first and last two theorems are for vector systems; the other theorems are for scalar equations. Theorem Consider the initial value problem: dx/dt = F(t, x) with x(t0 ) = x0 , wherenx = o x(t) = [x1 (t) x2 (t) . . . xn (t)]T . If each of the functions ∂Fi are continuous in a region R of (t, x) space containing {Fi } and ∂x j the point x0 , then there is an interval |t − t0 | < h in which there exists a unique solution to the problem.

Theorem Consider the initial value problem: y 0 = f (x, y) with y(x0 ) = y0 . Let the functions f be continuous in some rectangle a < x < b, c < y < d containing the point (x0 , y0 ). Assume that f (x, y) satisfies a Lipschitz condition in y. Then, in some interval x0 − h < x < x0 + h contained in a < x < b, there is a unique solution to the given problem. Theorem Consider the initial value problem: y 0 = f (x, y) with y(x0 ) = y0 . Let the functions f and ∂f /∂y be continuous in some rectangle a < x < b, c < y < d containing the point (x0 , y0 ). Then, in some interval x0 − h < x < x0 + h contained in a < x < b, there is a unique solution to the given problem. Theorem Consider the initial value problem: y 00 = f (x, y, y 0 ) with y(x0 ) = y0 , y 0 (x0 ) = y00 . Let the functions f , fy , and fy0 be continuous in an open region R of three-dimensional (x, y, y 0 ) space. If the point c CD-ROM Handbook of Differential Equations Academic Press 1997

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(x0 , y0 , y00 ) is in R, then there exists some interval about x0 for which there is a unique solution to the given problem. Theorem Consider the initial value problem: y (n) + p1 (x)y (n−1) + · · · + pn−1 (x)y 0 + p1 (x)y = q(x), with y(x0 ) = y0 ,

y 0 (x0 ) = y00 ,

(n−1)

y (n−1) (x0 ) = y0

...

.

If the functions {pi (x)} and q(x) are continuous on the open interval a < x < b, then there exists a unique solution to the problem. Theorem Consider the initial value problem: x0 = f (x, y, t),

y 0 = g(x, y, t)

with x(t0 ) = x0 , y(t0 ) = y0 . If f and g satisfy a Lipschitz condition (with respect to x and y) in the region {|t − t0 | ≤ A, |x − x0 | ≤ B, |y −y0| ≤ C}, then the problem has a unique solution in some interval a < t < b about the point t0 . Theorem Consider the boundary value problem: x00 = f (t, x, x0 ), x(0) = A,

0 < t < 1,

x(1) = B.

If f and fx are continuous and fx ≥ 0, then there exists a unique solution. Theorem Consider the initial value problem y 00 + f (x, y, y 0 ) = 0, B1 [y] = y 0 (a) + Ay(a) − C1 = 0,

(10.1)

0

B2 [y] = y (b) + By(b) − C2 = 0, where f satisfies a Lipschitz condition, and fy and fy0 are bounded for x in the interval [a, b] and for values of (y, y 0 ) of interest. Consider the two comparison equations u001 + h1 (x, u1 , u01 ) = 0, B1 [u1 ] = 0, u002 + h2 (x, u2 , u02 ) = 0, B1 [u2 ] = 0,

B2 [u1 ] = 0, B2 [u2 ] = 0,

with h1 (x, y, y 0 ) ≤ f (x, y, y 0 ) ≤ h2 (x, y, y 0 ). We assume that the u1 and u2 problems have unique solutions. Then there exists at least one solution to (10.1) in the given region, and every solution has the property u1 (x) ≤ y(x) ≤ u2 (x). (This theorem is one of the major results of the theory of differential inequalities.) c CD-ROM Handbook of Differential Equations Academic Press 1997

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 Cauchy–Kowalewski Theorem If the vector u = u1 satisfies u(0, x) = h(x), ut = A(u)ux ,

u2

...

un

T

where uk = uk (x, t), A(u) is an analytic matrix, and h(x) is an analytic function, then a neighborhood of t = 0 can be found in which there is a unique solution u, with each uk being analytic.

Example 1 The first order initial value problem y 0 = |y|1/3 ,

y(x0 ) = 0

(10.2)

has a right-hand side that is not Lipschitz continuous at y = 0. This equation, in fact, has an infinite number of solutions. Let x1 and x2 be any two numbers such that x1 < x0 < x2 . Then the following function 
2 3/2  (x1 − x)3/2 , if x < x1 , − 3 f (x) = 0, if x1 < x < x2 ,   2
3/2 3/2 (x − x2 ) , if x2 < x, 3 is a solution to equation (10.2).

Example 2 The nonlinear second order equation  0 3 + 24(1 − u) = 0, u(0) = 1, u0

u0 (0) = 0,

has at least three solutions: u(t) = 1, u(t) = 1 − t2 , and u(t) = 1 + t2 .

Notes 1. Differential equations with discontinuities (see page 264) and delay equations (see page 253) do not meet the requirements of the above theorems. They must be investigated separately. 2. It is often possible to determine when a linear ordinary differential equation has a unique solution. When the solution is not unique, it is sometimes possible to describe the degrees of freedom that make it non-unique using alternative theorems (see page 15). 3. Fixed point theorems are a specific method that can be used to prove the existence of a solution (see page 58). The section on well posed differential equations contains some results on existence and uniqueness (see page 115). 4. Bobisud and O’Regan [2] consider existence questions for some second order initial value problems of the form y 00 +F (t, y, y 0 ) = 0, where F is allowed to be suitably singular. For example, F (t, y, y 0 ) = t−1/2 y −1/2 is allowed. c CD-ROM Handbook of Differential Equations Academic Press 1997

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5. The existence of solutions to a differential equation can be critically dependent on the size of the coefficients in the equation. For example, Coddington and Levinson [3] show that the problem y 00 = −y 0 − (y 0 )3 , y(0) = A, y(1) = B

(A 6= B)

does not have a solution for small enough  > 0. 6. The classical problem −∇2 u = up

in Ω,

u=0

on ∂Ω,

where Ω is a bounded domain in RN , with smooth boundary ∂Ω, has the interesting existence property (see Peletier [8]): N +2 • If p < N −2 , then existence of a solution is assured for any domain Ω; N +2 • If p ≥ N −2 , then there exists no solution in any star-shaped domain.

Similar results are available for the equation ut = ∇2 u + up ; existence of a global positive solution depends on whether p is greater than 1 + 2/N (see Fujita [5]). 7. A classic result of Lewy [7] is that the equation −ux − iuy + 2(ix − y)uz = F (x, y, z), where F (x, y, z) is of class C ∞ , has no H 1 -solution, no matter what open (x, y, z) set is taken as the domain of existence. 8. Waterhouse [11] has the theorem: Theorem: Consider the homogeneous linear differential equation involving only derivatives of even order
and even functions as coefficients, D2n + a1 D2n−2 + · · · + an f = 0 with ai (x) = ai (−x) and having the symmetric homogeneous boundary conditions B1 (D)f (s) = · · · = Bn (D)f P (s) = 0 = B1 (D)f (−s) = · · · = Bn (D)f (−s) with Bi (D) = j bij Dj . If this boundary value problem has a non-trivial solution, and if each of the vectors (bi0 − bi1 , bi2 − bi3 , . . . ) is in the span of the vectors (b10 , b11 , b12 , . . . ) and (b20 , b21 , b22 , . . . ), then this problem has a nontrivial solution that is either even or odd. 9. Agarwal and Sheng [1] provide necessary and sufficient conditions for the existence and uniqueness of solutions of general nth order nonlinear differential equations satisfying Abel–Gontscharoff boundary conditions. These are boundary conditions of the form y (i) (ai+1 ) = Ai+1 for 0 ≤ i ≤ n−1 where −∞ < a ≤ a1 ≤ a2 ≤ · · · ≤ an ≤ b < ∞.

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References [1] Agarwal, R. P., and Sheng, Q. Abel–Gontscharoff boundary value problems. Math. Comput. Modelling 17, 7 (1993), 37–55. [2] Bobisud, L. E., and O’Regan, D. Existence of solutions to some singular initial value problems. J. Math. Anal. Appl. 133 (1988), 214–230. [3] Coddington, E. A., and Levinson, N. A boundary value problem for a nonlinear differential equation with a small parameter. Proc. Amer. Math. Soc. 3 (1952), 73–81. [4] Feckan, M. A new method for the existence of solution of nonlinear differential equations. J. Differential Equations 89 (1991), 203–223. [5] Fujita, H. On the blowing up of solutions of the Cauchy problem for zzzref18refzzz. J. Fac. Sci. Univ. Tokyo Sect. A. Math 16 (1966), 105–113. [6] Levine, H. A. The role of critical exponents in blowup theorems. SIAM Review 32, 2 (1990), 262–288. [7] Lewy, H. An example of a smooth linear partial differential equation without solution. Annals of Math. 66, 1 (July 1957), 155–158. [8] Peletier, L. A. Elliptic equations with nearly critical growth. In Equadiff 1987, C. M. Dafermos, G. Ladas, and G. Papanicolaou, Eds., no. 118 in Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York, 1987, pp. 561–574. [9] Plum, M. Computer-assisted existence proofs for two-point boundary value problems. Computing 46 (1991), 19–34. [10] Redheffer, R. Differential Equations. Jones and Bartlett Publishers, Boston, 1991. [11] Waterhouse, W. C. Some boundary value problems with even or odd solutions. SIAM Review 38, 4 (December 1996), 645–646.

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Definitions and Concepts

Fixed Point Existence Theorems

Applicable to

Differential equations of all types.

Yields A statement about the existence of the solution.

Idea If the statement concerning the existence of a solution to a differential equation can be interpreted as a statement concerning fixed points in a Banach space, then a fixed point theorem might be useful.

Procedure The Schrauder fixed point theorem states: Let X be a non-empty convex set in a Banach space and let Y be a compact subset of X. Suppose Y = f (X) maps X continuously into Y . Then there is a fixed point x∗ = f (x∗ ). By interpreting a given differential equation as a continuous function in a Banach space, the above theorem indicates the existence of a solution.

Example Suppose we wish to determine whether a solution exists to the nonlinear boundary value problem u00 = −e−u(x) , u(0) = u(1) = 0,

(11.1)

on the interval x ∈ [0, 1]. We first note that the problem v 00 = −φ(x), v(0) = v(1) = 0, has the solution

Z

1

G(x, z)φ(z) dz,

v(x) = 0

where G(x, z) is the Green’s function (see page 321) ( (1 − x)z, for 0 ≤ z ≤ x, G(x, z) = (1 − z)x, for x ≤ z ≤ 1. Hence, we can write equation (11.1) in the form of an equivalent integral equation Z 1 G(x, z)e−u(z) dz. (11.2) u(x) = f (u(x)) ≡ 0

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To apply Schrauder’s fixed point theorem to equation (11.2), we need to carefully define the Banach space B and the sets X and Y . If we define B = space of continuous functions on (0, 1), X = {u(x) | 0 ≤ u(x) ≤ 1, u(x) is continuous}, Y = f (X), then we can apply the theorem. Note that in this example, X is not compact but Y is. Note also that the bounds in X were derived after some analysis of equation (11.1). Finally, then, we conclude that equation (11.1) has a solution.

Notes 1. In the example above we used a fairly standard linearization trick that can be described in more generality. Suppose that an expression D(f, g) (which could involve derivatives of f and/or g) is linear in f . Suppose also that the linear differential equation D(f, g) = 0 has a unique solution f = T [g] for each g in some function space. Then to find a solution, in that function space, of the (possibly nonlinear) equation D(f, f ) = 0 is equivalent to finding a fixed point of the mapping T . Thus a particular nonlinear differential equation can be studied by means of a more general linear differential equation, together with a fixed point problem. 2. Once a differential equation has been formulated as a fixed point statement, numerical methods that search for fixed points in a function space can be used. See, for example, Allgower [1]. 3. Interval techniques (see page 545) may also be used to bound the solution of a fixed point statement. See Moore [7, Chapter 15, pages 97–102] for details. 4. A contraction mapping is a functional iteration, say yn+1 = N [yn ], that converges to the solution of the fixed point equation y = F [y]. The Picard iteration (see page 618) is such a mapping. 5. Another fixed point theorem that is of use in differential equations is Krasnoselskii’s theorem (see Franklin [3] for details): Consider the fixed point equation x = f(x) + g(x) for x in a Banach space B. Let X be a non-empty closed convex set in B. Let f(x) map X continuously into a compact subset Y ⊂ X. Let g(x) be a contraction mapping on X (note that the range of g need not be compact). If it is assumed that y + g(x) ∈ X for y ∈ Y and x ∈ X, then there is a fixed point of x = f(x) + g(x). 6. Another fixed point theorem that is of use in differential equations is the Tihonov fixed point theorem (see Iyanaga and Kawada [6, pages 542–543] for details): c CD-ROM Handbook of Differential Equations Academic Press 1997

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Let R be a locally compact topological linear space, A a compact convex subset of R, and T a continuous mapping sending A into itself. Then T has fixed points. 7. Existence theorems for solutions for differential equations may be found on page 53. 8. See also Burton [2, Chapter 3, pages 164–196], Hale [4, Appendix, pages 171–172], Hartman [5, Chapter 12, pages 404–449], Smart [8, Chapter 6, pages 41–52], and Stakgold [9, pages 243–259].

References [1] Allgower, E. L. Application of a fixed point search algorithm to nonlinear boundary value problems having several solutions. In Fixed Points: Algorithms and Applications, S. Karamardian, Ed. Academic Press, New York, 1977. [2] Burton, T. A. Perturbation and delays in differential equations. SIAM J. Appl. Math. 29, 3 (November 1975), 422–438. [3] Franklin, J. Methods of Mathematical Economics. Springer–Verlag, New York, 1980. [4] Hale, J. K. Oscillations in Nonlinear Systems. McGraw–Hill Book Company, New York, 1963. [5] Hartman, P. Ordinary Differential Equations. John Wiley & Sons, New York, 1964. [6] Iyanaga, S., and Kawada, Y. Encyclopedic Dictionary of Mathematics. MIT Press, Cambridge, MA, 1980. [7] Moore, R. E. Interval Analysis. Prentice–Hall, Inc., Englewood Cliffs, NJ, 1966. [8] Smart, D. R. Fixed Point Theorems. Cambridge University Press, New York, 1974. [9] Stakgold, I. Green’s Functions and Boundary Value Problems. John Wiley & Sons, New York, 1979.

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Hamilton–Jacobi Theory

12.

61

Hamilton–Jacobi Theory

Applicable to

Conservative dynamical systems.

Yields A reformulation of a system of ordinary differential equations.

Idea A change of variables may lead to more tractable equations.

Procedure A conservative dynamical system has a Lagrangian L defined by L = T − V , where T (V ) is the kinetic (potential) energy. If the generalized coordinates in this system are q = (q1 , q2 , . . . , qn ), then the equations of motion are given by   ∂L d ∂L − = 0, for i = 1, 2, . . . , n, (12.1) dt ∂ q˙i ∂qi where a dot denotes differentiation with respect to t. The equations in (12.1) are called Lagrange’s equations. If we define the generalized mo∂L menta by pi = ∂q and the Hamiltonian by H = pT q˙ − L, then Lagrange’s i equations become ∂H , ∂pi ∂H , p˙ i = − ∂qi ∂H ∂L =− . ∂t ∂t q˙i =

(12.2)

These equations are called Hamilton’s equations. If we change from the (H, p, q) variables to the (J, P, Q) variables via the canonical transformation defined by the generating function S(P, q, t) (see page 132), then ∂S , ∂qi ∂S , Qi = ∂Pi   ∂S . J(P, Q, t) = H p(P, Q, t), q(P, Q, t), t + ∂t pi =

(12.3)

In these new variables, Hamilton’s equations may be written ∂J , ∂Pi ∂J . P˙i = − ∂Qi

Q˙ i =

(12.4)

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If the canonical transformation is chosen so that J = 0, then (12.4) says that P and Q are constant. To have J vanish identically, we require (from (12.3))   ∂S ∂S ∂S ∂S H = 0. , ,..., , q1 , q2 , . . . , qn , t + ∂q1 ∂q2 ∂qn ∂t This last equation is known as the Hamilton–Jacobi equation. The procedure is to solve the Hamilton–Jacobi equation for the generating function S, make a canonical change of variables using this generating function, and then solve Hamilton’s equation in these new coordinates. This will yield a solution to Lagrange’s equations.

Example Suppose we want to solve the linear constant coefficient ordinary differential equation q¨ + ω 2 q = 0.

(12.5)


This differential equation comes from the Hamiltonian H = 12 p2 + ω 2 q 2 , which, in turn, corresponds to the following Hamilton–Jacobi equation: "  # 2 ∂S 1 ∂S 2 2 = 0. (12.6) +ω q + 2 ∂q ∂t To solve for S(q, t), we use separation of variables (see page 487), and look for a solution in the form S(q, t) = a(q) + b(t), for some unknown functions a(q) and b(t). Using this form for S in equation (12.6) and making the usual argument about which terms must depend upon which variables, we determine that a(q) and b(t) must satisfy  b˙ = −α,

da dq

2 + ω 2 q 2 = 2α,

Rp 2α − ω 2 q 2 dq. If where α is a separation constant. Hence, S = −αt + we call α = P , then we can compute from equation (12.3) Q=

∂S = −t + ∂P

Z

  1 ωq sin−1 √ , ω 2P h i sin ω(t+Q) , which is the solution

(2P − ω 2 q 2 )−1/2 dq = −t + √

which may be inverted to yield q =

2P ω

to equation (12.5).

Notes 1. Lagrange’s equations can be interpreted asRthe variational or Euler– Lagrange equations for the functional J = L dt (see page 418). c CD-ROM Handbook of Differential Equations Academic Press 1997

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2. The functions f and g are said to be in involution or to Poisson commute if the Poisson bracket [f, g] is identically equal to zero. Liouville’s theorem states that a function F is a first integral of a system with Hamiltonian function H if and only if H and F are in involution. See Abraham et al. [1, page 471] for details. 3. Poisson’s theorem states that the Poisson bracket of two first integrals of a Hamiltonian system is again a first integral. See Goldstein [2, Chapter 9, pages 273–317] for details. 4. Any function A(p, q) defined along the trajectories of equation (12.2) satisfies  X  ∂A ∂H dA ∂A ∂H = [A, H] = − dt ∂qj ∂pj ∂pj ∂qj j where the square brackets denote the Poisson bracket. 5. A general form for a non-conservative system is often taken to be ∂C ∂D + ∂pi ∂qi ∂C ∂D p˙ i = − + ∂qi ∂pi q˙i =

(12.7)

Where C(p, q) and D(p, q) are called the conservative and dissipation functions. For D = 0, this reduces to equation (12.2). For C = 0, this becomes a gradient system. Any function A(p, q) defined along the trajectories of equation (12.7) satisfies dA = ∇ A · ∇ D + [A, C]. dt Choosing A = C and A = D, we obtain the evolution equations for the conservative and dissipative functions dC = ∇ C · ∇ D, dt dD = ∇ D · ∇ D + [D, C]. dt Note that ∇2 D equals the divergence of the vector field of equation (12.7) and that the system is dissipative when ∇2 D < 0. ˙ t), the inverse problem 6. Given the equations of motion: q¨i = fi (q, q, of classical mechanics is to determine whether these equations are equivalent to the Euler–Lagrange equations based on a Lagrangian L. ˙ t) is desired so that That is, a matrix w = w(q, q, d qj − fj ) = wij (¨ dt



∂L ∂ q˙i

 −

∂L . ∂qi

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The necessary and sufficient conditions for the existence of w and L are called the Helmholtz conditions, they are ∂wij ∂wik = , wij = wji , ∂ x˙ k ∂ x˙ j 1 1 ∂fk ∂fk Dwij = − wik − wjk 2 ∂ x˙ j 2 ∂ x˙ i   1 ∂fk ∂fk ∂fk ∂fk D wik = wik − wjk − wjk 2 ∂ x˙ j ∂ x˙ i ∂xj ∂xi  P  ∂ with D = ∂t + m x˙ m ∂x∂m + fm ∂ x∂˙ m . See Hojman and Shepley [4]. 7. The KdV equation, ut = −uxxx +6uux, can be treated as a Hamiltonian system, ut = {u, H}, with the Hamiltonian and Poisson brackets defined by Z   1 u2 (x) dx {u(x), u(y)} = −∂ 3 + 4u∂ + 2ux δ(x − y) H= 2 8. See also Haar [3, Chapter 6, pages 121–145] and Nayfeh [5, pages 179–189].

References [1] Abraham, R., Marsden, J. E., and Ratiu, T. Manifolds, Tensor Analysis, and Applications. Addison–Wesley Publishing Co., Reading, MA, 1983. [2] Goldstein, H. Classical Mechanics. Addison–Wesley Publishing Co., Reading, MA, 1950. [3] Haar, D. Elements of Hamiltonian Mechanics. Pergamon Press, New York, 1971. [4] Hojman, S. A., and Shepley, L. C. No Lagrangian? No quantization! J. Math. Physics 32, 1 (Jan 1991), 142–146. [5] Nayfeh, A. H. Perturbation Methods. John Wiley & Sons, New York, 1973. [6] Sanz-Serna, J. M. Runge–Kutta schemes for Hamiltonian systems. BIT 28 (1988), 877–883.

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Integrability of Systems

13.

65

Integrability of Systems

Applicable to

Systems of differential equations.

Yields Information about whether a Hamiltonian system is completely integrable.

Idea The Painlev´e test performs a singular point analysis, which gives information about integrability.

Procedure An autonomous Hamiltonian system is called (Liouville) integrable if there exists another function I such that [H, I] = 0. This function must be functionally independent of H, it must exist globally and be single valued, and it must be a complex analytic function of its variables. If the Hamiltonian system has N degrees of freedom it is called completely integrable if it possesses N independent single valued analytic first integrals {Ik } that commute with respect to the Poisson bracket [In , Im ] =

 N  X ∂In ∂Im ∂In ∂Im = 0. − ∂qi ∂pi ∂pi ∂qi i=1

One of these first integrals will be the Hamiltonian itself. Given a Hamiltonian system, there is no known systematic method for determining whether or not that system is integrable. Much recent work has focused on the Painlev´e test. The test asserts that an equation is integrable if every ordinary differential equation that arises as a similarity reduction of an integrable partial differential equation has the Painlev´e property; that is, it has no movable singularities except poles, perhaps after a transformation of variables. For the Painlev´e test to be effective, it is necessary to determine the complete symmetry group of the differential equation under consideration. If it passes the test, then it is believed that the original partial differential equation will be solvable by inverse scattering methods (see page 460). The Painlev´e test also has applications in determining the stability of systems of ordinary differential equations. Roughly speaking, a partial differential equation is said to possess the Painlev´e property if the only singularities of the general solution on arbitrary non-characteristic surfaces are poles. Singular point analysis is used to determine if differential equations have the Painlev´e property. The test consists of substituting u(x) =

∞ X

un (x − x0 )α+p ,

n=0

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for α < 0, into the tested equation in the vicinity of a singular point x0 and investigating whether this expansion is compatible with the equation and contains a sufficient number of undetermined coefficients for the approximation of a general solution.

Example The motion of the N particle lattice is described by the Hamiltonian H(p, q) =

N N 1 X 2 X qj −qj+1 pj + e 2 j=1 j=1

(13.1)

where qN +1 = q1 (which corresponds to cyclic boundary conditions). If {aj , bj } are defined by aj :=

1 (qj −qj+1 )/2 e , 2

bj :=

1 pj , 2

then the equations of motion are a0j = aj (bj − bj+1 ),

b0j = 2(a2j−1 − a2j ).

(13.2)

If the following N × N matrices are defined:   b 1 a1 0 . . . 0 aN  a1 b 2 a2 0 0     0 a2 b 3 0 0    L= .  ..   .. .   0 0 0 bN −1 aN −1  aN 0 0 aN −1 bN   0 −a1 0 ... 0 aN  a1 0 −a2 0 0     0 a2 0 0 0    A= . , .. ..  ..  . .    0 0 0 ... 0 −aN −1  −aN 0 0 aN −1 0 then equation (13.2) may be written in the form dL = [A, L] = AL − LA. dt d(Lk ) = [A, Lk ] for any positive integer k. From dt this it follows that the trace of the matrix Lk is constant. Hence, the traces { tr (L), tr (L2 ), . . . , tr (Lk ), . . . } are first integrals for (13.2). They turn out to be independent and in involution of each other. Hence, the Hamiltonian in equation (13.1) is completely integrable.

Note we also have

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Notes 1. Several definitions of “integrability” are in use in the literature. For example, the PDE N (x, t, u) = 0 with u(x, 0) = f (x) is called completely integrable if there is an integral equation for K of the form Z ∞ K(x, y; t) + F (x, y, ; t) + K(x, z; t)H(z, y; t) dz = 0 x

called the Gelfand–Levitan equation, such that • F and H are uniquely determined from f (x) • the solution of the PDE is given by u(x, t) = K(x, x; t). 2. Completely integrable PDEs are known to possess several remarkable properties including: • the existence of soliton solutions (see page 626) • the existence of an infinite number of independent conservation laws (see page 47) • a Lax representation (see page 460) • B¨acklund transformations (see page 428) 3. In general, linear equations only have fixed singularities while nonlinear equations can have both fixed and movable singularities. • Consider the linear equation y 00 + p(x)y 0 + q(x)y = 0 which has the general solution y(x) = Ay1 (x) + By2 (x) where A and B are arbitrary constants. The location of the singularities of y(x) depend only on p(x) and q(x), not on A or B. The singularities of this equation are fixed, since they do not depend upon the constants of integration. • Consider the nonlinear equation y 0 +y 2 = 0 which has the general solution y(x) = (x − x0 )−1 where x0 is an arbitrary constant. In this case y(x) has a singularity, a pole, which is movable since it depends on the constant of integration x0 . 4. For first order equations of the form y 0 = F (y, x), where F is rational in y and analytic in x, the only equation which has no movable singularities other than poles is the Riccati equation y 0 = p0 (x) + p1 (x)y + p2 (x)y 2 . For second order equations of the form y 00 = F (y, y 0 , x), where F is rational in y and y 0 and analytic in x, Painlev´e et al. (see Ince [8]) showed that there are only 50 canonical equations which have no movable singularities except poles. Of these, 44 are integrable in terms of known functions (such as elliptic functions) and the remaining 6 defined new transcendental functions, called the Painlev´e transcendents (see page 128). p2 +p2 +p2 5. The three-particle Toda lattice has the Hamiltonian H = 1 22 3 + V with the potential energy V = ep1 −p2 + ep2 −p3 + ep3 −p1 . The sysc CD-ROM Handbook of Differential Equations Academic Press 1997

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Parameters

Invariant

b = 2σ

(x2 − 2σz)e2σt   1 2 3 −rx2 + y 2 + xy + x2 z − x4 e4t/3 3 3 4
2t 2 2 y +z e   1 4(1 − r)z + rx2 + y 2 − 2xy + x2 z − x4 e4t 4
2t 2 2 2 −rx + y + z e   1 4 4σt (2σ − 1)2 2 x + y 2 − (4σ − 2)xy + x2 z − x e σ 4σ

b = 0, σ =

1 3

b = 1, r = 0 b = 4, σ = 1 b = 1, σ = 1 b = 6σ − 2, r = 2σ − 1

Table 13.1: First integrals for the Lorenz equations.

tems admits three independent integrals, for instance, the functions I1 = p1 + p2 + p3 I2 = p1 p2 + p2 p3 + p3 p1 − V I3 = p1 p2 p3 − p1 ep2 −p3 − p2 ep3 −p1 − p3 ep1 −p2 These integrals are in involution and they are independent. 6. Consider the Hamiltonian H = (p2x + p2y )/2 +V: • For the H´enon–Heiles potential V = µ3 y 3 + x2 y, the system is integrable for µ = 1, 6, and 16. • For the Holt potential V = µ3 y 4/3 + x2 y −2/3 , the system is integrable for µ = 1, 6, and 16. • For the quartic potential V = ax4 + bx2 y 2 + cy 4 , the system is integrable if a : b : c have the ratios a:0:c, 1:2:1, 1:6:1, 1:12:16, 16:12:1, 1:6:8, or 8:6:1. 7. The Lorenz equations (see page 199) x0 = σ(y − x) y 0 = −y − xz − rx z 0 = xz − bz have known first integrals for several possible values of the parameters {σ, r, b}. For example, the first integrals in table 13.1 are known. 8. Clarkson et al. [4] state that the only third-order semilinear partial differential equations that are linearizable are equivalent to the c CD-ROM Handbook of Differential Equations Academic Press 1997

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following six equations: ut = uxxx + γux , ut = uxxx + uux + γux , ut = uxxx + u2 ux + γux ,
1 ut = uxxx − u3x + αeu + βe−u ux + γux , 8
−1 3 3 − P (u)(u2x + 1)ux + γux , ut = uxxx − ux u2xx 1 + u2x 2 2 3 −1 2 3 −1 ut = uxxx − ux uxx + αux − P (u)u2x + γux, 2 2 where P (u) is the Weierstrass elliptic function and satisfies 

dP du

2 = 4P 3 − δP − .

9. Clarkson et al. [4, page 1205] show that the PDE • ut = uxx + h(u)ux , where h(u) is a rational function of u, can pass the Painlev´e test only if h(u) is a linear function of u. • ut = uxxx + (uuxx + u2x ) + 32 (α − 1)u2 ux , where α is a constant, can pass the Painlev´e test only if α = 0, 3/2, or 3. 10. Hereman and Angenent [7] and Rand and Winternitz [13] describe Macsyma programs for determining whether a nonlinear ordinary differential equation has the Painlev´e property. (The differential equation must be a polynomial in both the dependent and independent variables and in all derivatives.) 11. The only equations of the form uxt = f (u), where f (u) is a linear combination of exponentials, which pass the Painlev´e test are: the sine–Gordan equation uxt = sin u, the Liouville equation uxt = eu , and the Bullough–Dodd equation uxt = eu − e−2u . 12. Polynomial potentials arise in many problems, particularly when truncated Taylor series are used to facilitate analytical study. It is useful to examine the integrability of such potentials. In two dimensions there are only three independent integrable cubic potentials; they are x3 + 3xy 2 + αy 3 , 2x3 + xy 2 , and 16x3 + 3xy 2 ; see Cleary [5]. 13. The Mathematica package DSolveIntegrals can compute complete integrals of non-linear PDEs. For example, given yuy = u + x2 u2x the integral is determined to be u = (−a2 + 4by − 2a log x − log2 x)/4. 14. The following equations are known to be completely integrable: sine– Gordon equation, Doff–Bullough, Ernst equation, axisymmetric stationary Einstein–Maxwell equation.

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References [1] Albrecht, D. W., Mansfield, E. L., and Milne, A. E. Algorithms for special integrals of ordinary differential equations. J. Phys. A: Math. Gen. 29 (1996), 973–991. [2] Buchler, J. R., Ipser, J. R., and Williams, C. A., Eds. Integrability in Dynamical Systems. New York Academy of Sciences, New York, 1988. [3] Chang, Y. F., Tabor, M., and Weiss, J. Analytic structure of the Henon–Heiles Hamiltonian in integrable and nonintegrable regimes. J. Math. Physics 23, 4 (April 1982), 531–538. [4] Clarkson, P. A., Fokas, A. S., and Ablowitz, M. J. Hodograph transformations of linearizable partial differential equations. SIAM J. Appl. Math. 49, 4 (August 1989), 1188–1209. [5] Cleary, P. W. Nonexistence and existence of various order integrals for two- and three-dimensional polynomial potentials. J. Math. Physics 31, 6 (June 1990), 1351–1355. [6] Gerdt, V. P., Zharkov, A. Y., Svinolupov, S. I., and Shabat, A. B. The use of computer algebra to investigate the integrability of non-linear evolution systems. U.S.S.R. Comput. Maths. Math. Phys. 28, 6 (1988), 50– 57. [7] Hereman, W., and Angenent, S. The Painleve test for nonlinear ordinary and partial differential equations. Macsyma Newsletter (January 1989), 11– 18. [8] Ince, E. L. Ordinary Differential Equations. Dover Publications, Inc., New York, 1964. [9] Mel’nikov, V. K. New method for deriving nonlinear integrable systems. J. Math. Physics 31, 5 (May 1990), 1106–1113. [10] Prelle, M. J., and Singer, M. F. Elementary first integrals of differential equations. Trans. Amer. Math. Soc. 279, 1 (September 1983), 215–229. [11] Progrebkov, A. K. On the formulation of the Painleve test as a criterion of complete integrability of partial differential equations. Inverse Prob. 5 (1989), L7–L10. [12] Ramani, A., Dorizzi, B., Grammaticos, B., and Bountis, T. Integrability and the Painleve property for low-dimensional systems. J. Math. Physics 25, 4 (April 1984), 878–883. [13] Rand, D. W., and Winternitz, P. ODEPAINLEVE — a Macsyma package for Painleve analysis of ordinary differential equations. Comput. Physics Comm. 42 (1986), 359–383. [14] Roekaerts, D., and Schwarz, F. Painleve analysis, Yoshida’s theorems and the direct methods in the search for integrable Hamiltonians. J. Phys. A: Math. Gen. 20 (1987), L127–L133.

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Internet Resources

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Internet Resources

Applicable to

Many topics related to differential equations.

Procedure Much information about differential equations is available through the internet. We list next some of these resources:

Symbolic software packages 1. There are a multitude of commercial computer packages available for symbolically solving differential equations (see page 240). These include • • • • • •

AXIOM Derive Macsyma Maple Mathematica REDUCE

http://www.nag.co.uk:80 http://www.derive.com http://www.macsyma.com http://www.maplesoft.com http://www.wolfram.com http://www.rrz.uni-koeln.de/REDUCE

2. The program CONVODE will symbolically solve ordinary and partial differential equations across the internet. For example, sending depend y,x; CONVODE( {df(y,x,2)+4*y=0}, {y}, {x}, {}, {english});

to [email protected] will have the solution of y 00 + 4y = 0 returned via email with comments in English (the default is French). See http://www.physique.fundp.ac.be/physdpt/ administration/convode.html. 3. MathServ provides an interface between the user and Mathematica (a symbolic computational engine). Templates for twelve different types of ODEs are available; the user can specifiy the functions appearing in them. The results are returned directly to your browser. See http://math.vanderbilt.edu/~pscrooke/detoolkit.html.

Numerical software packages There are a multitude of commercial computer packages available for numerically solving differential equations (see page 654). In particular, the Guide to Available Mathematical Software (GAMS) has a taxonomy of software classes, with many representatives of most classes. See http:// gams.nist.gov. This section lists a few packages that currently may be used freely for non-commercial purposes. • Diffpack is a collection of C++ class libraries aimed at the numerical solution of partial differential equations. The Diffpack home page c CD-ROM Handbook of Differential Equations Academic Press 1997

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int main() { real r0=.5, r1=1., x0=0., y0=0., theta0=0., theta1=1.; // parameters AnnulusMapping annulus(r0,r1,x0,y0,theta0,theta1); // annulus mapping MappedGrid mg(annulus); // MappedGrid for an annulus mg.update(); // create default variables realMappedGridFunction u(mg); // declare grid function on the grid u=1.; // initial condition u=1 MappedGridOperators op(mg); // difference operators and BCS u.setOperators(op); // associate with a grid function real t=0, dt=.005, a=1., b=1., nu=.1; // problem parameters for( int step=0; step<100; step++ ) { // loop for number of time steps u.display("solution"); // print out the solution u+=dt*((-a)*u.x()+(-b)*u.y()+nu*(u.xx()+u.yy())); // forward Euler step t+=dt; u.applyBoundaryCondition(0,BCTypes::dirichlet,BCTypes::allBoundaries,0.); // apply Boundary condition u=0 u.finishBoundaryConditions(); // fix up corners, periodic update } return 0; }

Program 14.1: Overture program for a reaction diffusion problem

is http://www.oslo.sintef.no/avd/33/3340/diffpack. The code can be downloaded from http://www.oslo.sintef.no/diffpack/ pub1.4 or from Netlib at http://www.netlib.org. • DsTool is A Dynamical System Toolkit with an Interactive Graphical Interface. It computes Poincar´e sections and bifurcation diagrams and is easily extensible. It was created at Cornell University and runs under X windows. The program and documentation can be obtained via ftp from macomb.cam.cornell.edu in the /pub/dstool directory. • KASKADE is a C++ package that solves elliptic partial differential equations. It is an adaptive multilevel-code for linear scalar elliptic and parabolic problems in 1, 2, and 3 space dimensions. It includes examples for nonlinear methods used in obstacle, porous media, and Stefan problems. It can be obtained via ftp from elib.zib-berlin.de in the directories /pub/kaskade/3.x and /pub/kaskade/Manuals/3.0. • Overture is a high level object oriented framework for solving PDEs on structured grids and overlapping grids using finite difference and finite volume methods. Overture is freely available and can be obtained from http://www.c3.lanl.gov/~henshaw/Overture/Overture.html. For example, the entire program to solve the problem ut +aux +buy = ν(uxx + uyy ) in an annulus A, with u(t = 0, A) = 1 and u ∂A = 0, using forward Euler’s method, is in program 14.1. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Electronic journals The Electronic Journal of Differential of Equations (EJDE) is dedicated to the rapid dissemination of high quality research in mathematics. Publications are available as PostScript, TEX, and DVI files. All topics related to differential equations and their applications are considered for publication. Research articles are refereed under the same standards as those used by the finest-quality printed journals. EJDE may be found at http://ejde.math.swt.edu.

Other resources • C*ODE*E is the acronym for the Consortium of ODE Experiments. Their goal is to share the rapidly growing wealth of computational instruction techniques with teachers of differential equations. The Consortium publishes a newsletter designed to provide a regular source of ideas, inspiration, and experiments for instructors of ODEs. The newsletter is available on-line and in print format. Their URL is http://www.math.hmc.edu/codee. • IDEA is the acronym for Internet Differential Equations Activities. This is an interdisciplinary effort to provide students and teachers with computer based activities for differential equations in a wide variety of discplines. This is sponsored by the NSF. It includes a glossary of terms and many other features. Their URL is http:// www.sci.wsu.edu/idea. • The American Mathematical Society maintains materials organized by mathematical subject classification at http://www.ams.org/mathweb/ mi-mathbyclass.html. In this classification, category 34 is “Ordinary differential equations” and category 35 is “Partial differential equations.” The AMS Preprint Server for these categories may be found at http://www.ams.org/preprints/34/msc34-page.html and http://www.ams.org/preprints/35/msc35-page.html. • Los Alamos maintains a web site on “Exactly Solvable and Integrable Systems”, see http://xxx.lanl.gov/archive/solv-int. • The Norwegian University of Science and Technology maintains a “Conservation Laws Preprint Server” at http://www.math.ntnu.no/ conservation. • The “Mathematics Archives,” see http://archives.math.utk.edu, is supported by the NSF, the State of Tennessee, Calvin College, and the University of Tennessee, Knoxville. Their repository of links related to ordinary differential equations and partial differential equations may be found at http://archives.math.utk.edu/topics/ ordinaryDiffEq.html and http://archives.math.utk.edu/topics/ partialDiffEq.html. c CD-ROM Handbook of Differential Equations Academic Press 1997

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• The Math/CS Department of Nebraska Wesleyan University has a differential equations resource page documenting course materials (labs and projects) developed as part of an NSF/ILI grant. The URL is http://brillig.nebrwesleyan.edu/delabs. • The Math Department at Oregon State University has developed a web-based study guide for several of its courses. The URL for the ODE home page is http://iq.orst.edu.mathsg/ode/ode.html.

Note 1. The URLs in this section are subject to change.

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Inverse Problems

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Inverse Problems

Applicable to

Inverse problems.

Yields Information about parameters appearing in a differential equation.

Idea There are theorems that can be used to determine which inverse problems may be solved.

Procedure The field of inverse problems is filled with specialized theorems that are useful for specific applications.

Example 1 Consider the eigenvalue problem −u00 + q(x)u = λu, u(0) cos α + u0 (0) sin α = 0,

for 0 ≤ x ≤ 1, (15.1)

0

u(1) cos β + u (1) sin β = 0, where λ is a complex parameter, q(x) is a real-valued function that is integrable on the interval [0, 1], and α and β are values in the interval [0, π). One common inverse problem consists of determining the function q(x) from the eigenvalues of equation (15.1). There are many different results in this area. For example Theorem Suppose that (α, β, q(x)) give rise to the eigenvalues {λn } and suppose that (α, β, q(x)) give rise to the eigenvalues {λn }. If λn = λn for n = 0, 1, . . . ; q(x) = q(x) for x ∈ (0, 12 ); and α = α, then q(x) = q(x) almost everywhere on the interval (0, 1). Another typical theorem is the following: Theorem Let λ0 < λ1 < λ2 < . . . be the eigenvalues of the problem −y 00 + q(x)y = λy with y 0 (0) = y 0 (π) = 0, where q(x) is a real-valued continuous function. If λn = n2 for n = 0, 1, 2, . . . , then q(x) = 0.

Example 2 One common technique to show uniqueness for an inverse problem is to investigate a mapping between the solutions of two equations with different values for the parameter(s) of interest. We have, for example (see Rundell [11]): c CD-ROM Handbook of Differential Equations Academic Press 1997

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Theorem Let u(x) and v(x) satisfy ut = uxx − a(x)u,

ux (0, t) = 0,

vt = vxx − a(x)v,

vx (0, t) = 0,

for 0 ≤ xR ≤ 1 and 0 ≤ t < T . If u(0, t) = v(0, t), then v(x, t) = x u(x, t)+ 0 K(x, s)u(s, t) ds, where K(x, s) satisfies the Goursat problem Kss − Ktt = (a(s) − a(x)) K(x, s), Ks (x, 0) = 0 Z x 1 K(x, x) = (a(r) − a(r)) dr 2 0 In this case it is possible to show that if positive function f (x), then a = a.

for 0 ≤ s ≤ x ≤ 1, for 0 ≤ x ≤ 1, for 0 ≤ x ≤ 1. Rx 0

K(x, s)f (s) ds = 0 for some

Notes 1. The numerical methods used to solve inverse problems tend to result in ill-conditioned systems. 2. If the spectra {λi } and {µi } are known for the following two problems ¯ (with H 6= H): −y 00 + q(x)y = λy, y 0 (0) − hy(0) = 0, y 0 (1) − Hy(1) = 0,

−y 00 + q(x)y = µy, y 0 (0) − hy(0) = 0, ¯ = 0, y 0 (1) − Hy(1)

¯ are all uniquely determined. See Rundell and then {q(x), h, H, H} Sacks [12].

References [1] Anger, G. Inverse Problems in Differential Equations. Plenum Publishing Corp., New York, 1990. [2] Barnes, D. C. The inverse eigenvalue problem with finite data. SIAM J. Math. Anal. 22, 3 (May 1991), 732–753. [3] Cannon, J. R., and Lin, Y. An inverse problem of finding a parameter in a semi-linear heat equation. J. Math. Anal. Appl. 145, 2 (1990), 470–484. [4] Castillo, R. D. R. On boundary conditions of an inverse Sturm–Liouville problem. SIAM J. Appl. Math. 60, 6 (December 1990), 1745–1751. [5] Colton, D., Ewing, R., and Rundell, W., Eds. Inverse Problems in Partial Differential Equations. SIAM, Philadelphia, PA, 1990. [6] Eskin, G. Inverse spectral problem for the Schroedinger equation with periodic vector potential. Comm. Math. Physics 125, 2 (1989), 263–300. [7] Hassan, A. A. M., and Abdel-Halim, I. H. Some inverse eigenvalue problems for the Laplacian operator, II. J. Inst. Math. Comput. Sci. Math. Ser. 2, 2 (1989), 125–146. c CD-ROM Handbook of Differential Equations Academic Press 1997

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[8] Levitan, B. M., and Sargsjan, I. S. Sturm–Liouville and Dirac Operators. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. [9] Pilant, M., and Rundell, W. Determining a coefficient in a first-order hyperbolic equation. SIAM J. Appl. Math. 51, 2 (April 1991), 494–506. [10] Roy, D. N. G. Methods of Inverse Problems in Physics. CRC, Boca Raton, FL, 1990. [11] Rundell, W. The use of integral operators in undetermined coefficient problems for partial differential equations. Appl. Analysis 18 (1984), 309– 324. [12] Rundell, W., and Sacks, P. E. Reconstruction techniques for classical inverse Sturm–Liouville problems. Math. of Comp. 58, 197 (January 1992), 161–183.

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Definitions and Concepts

Limit Cycles

Applicable to

Systems of nonlinear autonomous differential equa-

tions.

Yields Knowledge of whether or not there exist limit cycles.

Idea Knowing that limit cycles exist for a differential system allows global characterizations of the differential system.

Procedure A non-constant solution of the system dx dt = f(x) is called a cycle (or a limit cycle) if there is a positive number T (called the period of the cycle) such that x(t + T ) = x(t) for all t. It is easy to show that inside of every cycle is at least one critical point (i.e., a point where f(x) = 0, see page 526). In many systems it is not only true that there are finitely many cycles but also that all solutions tend to one of these cycles. This knowledge permits a concise characterization of the phase plane.

Example 1 The nonlinear autonomous system dx = −y + x(1 − x2 − y 2 ), dt dy = x + y(1 − x2 − y 2 ) dt becomes, under the change of variables {x = r cos θ, y = r sin θ}, the uncoupled system dθ dr = r(1 − r2 ), = 1. dt dt These new equations have the solution 1 r(t) = √ , 1 + Ae−2t

θ(t) = t + B,

where A and B are arbitrary constants. Hence, the solution of the original system is cos(t + B) sin(t + B) x(t) = √ , y(t) = √ . 1 + Be−2t 1 + Be−2t This states that all solutions tend to the circle x2 (t) + y 2 (t) = 1 as t → ∞. Of course, in most circumstances it is not possible to construct explicitly the limit cycle. Generally theorems (such as those below) are used to prove the existence of a limit cycle. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Example 2 The Van der Pol equation
dx d2 x +x=0 − µ 1 − x2 2 dt dt with µ > 0 has limit cycles. For this equation, there is negative damping for small values of x and positive damping for large values of x. Hence the value of x increases when x is small and it decreases when x is large.

Notes 1. Given a limit cycle Γ and a positive number a, define the annulus centered on Γ to be {x | distance from x to Γ is less than a} where the distance from x to Γ is defined to be min|x − u|. u∈Γ

A cycle Γ is called isolated if there is a positive number a for which the annulus centered on Γ contains no other limit cycles. A cycle is non-isolated if every annulus centered of Γ contains at least one other limit cycle. The system
dx = x sin x2 + y 2 − y, dt


dy = y sin x2 + y 2 + x dt

has infinitely many isolated cycles whereas the system {x0 = y, y 0 = −x} has infinitely many non-isolated cycles. 2. Part of Hilbert’s 16th problem asked for the maximum number of limit cycles of the system {x0 = A(x, y), y 0 = B(x, y)} where A and B are polynomials. If A and B are polynomials of degree n, then the maximum number is known as the Hilbert number or the Hilbert function, Hn . It is known that H0 = 0, H1 = 0, H2 ≥ 4, H3 ≥ 8, Hn ≥ n−1 2 if n is odd, and Hn < ∞. The example that demonstrates that H2 ≥ 4 (found by Songling [12]) is x0 = ax − y − 10x2 + (5 + b)xy + y 2 , y 0 = x + x2 + (8c − 25 − 9b)xy, where a = −10−200 , b = −10−13 , and c = −10−52 . See also James and Lloyd [4]. 3. Neto [8] has the two results: Theorem The equation x0 = a2 x2 +a1 x+a0 , where the {ai } are continuous functions on [0, 1], has at most two closed solutions, if not all solutions in [0, 1] are closed. and Theorem The equation x0 = a3 x3 + a2 x2 + a1 x + a0 , where the {ai } are continuous functions on [0, 1], has at most three closed solutions. c CD-ROM Handbook of Differential Equations Academic Press 1997

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4. If f (x) and g(x) are continuous, have continuous derivatives, and satisfy the conditions: • xg(x) > 0 for x 6= 0, • f (x) is negative in the interval a < x < b (with a < 0 and b > 0) and positive outside of this interval, R∞ R0 • 0 f (x) dx = −∞ f (x) dx = ∞, then every nontrivial solution of Li´enard’s equation dx d2 x + g(x) = 0 + f (x) dt2 dt

(16.1)

is either a limit cycle or a spiral that tends toward a limit cycle as t → ∞. See Birkhoff and Rota [1, pages 135–137] for details. 5. Li´enard’s theorem states If f (x) and g(x) are continuous and satisfy the conditions Rx • F (x) := 0 f (x) dx is an odd function, • F (x) is zero only at x = 0, x = a, x = −a, for some a > 0, • F (x) → ∞ monotonically for x > a, • g(x) is an odd function, and g(x) > 0 for x > 0, then equation (16.1) has a unique limit cycle. For details, see Jordan and Smith [5]. Note that Van der Pol’s equation (see example 2) satisfies Li´enard’s theorem and, hence, has a unique limit cycle. 6. Bendixson’s theorem states (see Simmons [11, pages 338–352]) ∂G If ∂F ∂x + ∂y is continuous and is always positive or always negative in a certain region of the phase plane, then the autonomous system dy dx = F (x, y), = G(x, y) dt dt has no limit cycles in that region.

For example, the equation for the Lewis regulator dx d2 x + x = 0, + (1 − |x|) dt2 dt which is equivalent to dx = F (x, y) = y, dt

dy = G(x, y) = −x − (1 − |x|)y, dt

∂G has ∂F ∂x + ∂y = |x| − 1. Hence, the Lewis regulator has no limit cycles in the strip −1 < x < 1. 7. Another statement of Bendixson’s theorem, regarding periodic solutions or limit cycles, can be stated as follows:

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Limit Cycles

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Consider x˙ = f (x) in a simply connected domain D (in two dimensions). If the gradient of f is not identically zero over any subregion of D and does not change sign in D, then D contains no closed trajectory. 8. The Levinson–Smith theorem states (see Hagedorn [3, page 143]) For the differential equation x00 + f (x, x0 )x0 + g(x) = 0

(16.2)

if the following conditions are satisfied: • • • •

> 0 for all x > 0, Rxg(x) ∞ g(x) dx = ∞, 0 f (0, 0) < 0, there exists an x0 > 0 such that f (x, x0 ) ≥ 0 for |x| > x0 , for every x0 , • there exists a constant M > 0, such that f (x, x0 ) ≥ −M for |x| ≤ x0 , Rx • there exists an x1 > x0 such that x01 f (x, v(x)) dx ≥ 10M x0 , where v(x) is any arbitrary positive and monotonically decreasing function of x,

then equation (16.2) has at least one limit cycle. 9. Sedaghat [9] shows that factorable planar systems (i.e., systems of the form x0 = f (x)h(y) and y 0 = k(x)g(y)) do not have limit cycles.

References [1] Birkhoff, G., and Rota, G.-C. Ordinary Differential Equations. John Wiley & Sons, New York, 1978. [2] Blows, T. R., and Lloyd, N. G. The number of limit cycles of certain polynomial differential equations. Proc. Roy. Soc. Edinburgh 98A (1984), 215–239. [3] Hagedorn, P. Non-linear Oscillations. Clarendon Press, Oxford, England, 1982. [4] James, E. M., and Lloyd, N. G. A cubic system with eight smallamplitude limit cycles. IMA J. Appl. Mathematics 47 (1991), 163–171. [5] Jordan, D. W., and Smith, P. Nonlinear Ordinary Differential Equations, second ed. Clarendon Press, Oxford, England, 1987. [6] Koditschek, D. E., and Narendra, K. S. Limit cycles of planar quadratic differential equations. J. Differential Equations 54 (1984), 181–195. [7] Kuang, Y. Finiteness of limit cycles in planar autonomous systems. Appl. Anal. 32, 3–4 (1989), 253–264. [8] Neto, A. L. On the number of solutions of the equation zzzref11refzzz, zzzref12refzzz, for which x(0)=x(1). Inventiones Mathematicae 59 (1980), 67–76. c CD-ROM Handbook of Differential Equations Academic Press 1997

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[9] Sedaghat, H. Geometric properties of factorable planar systems of differential equations. SIAM Review 38, 4 (December 1996), 660–665. [10] Shahshahani, S. Periodic solutions of polynomial first order differential equations. Nonlinear Analysis 5, 2 (1981), 157–165. [11] Simmons, G. F. Differential Equations with Applications and Historical Notes. McGraw–Hill Book Company, New York, 1972. [12] Songling, S. A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sinica 23 (1980), 153–158. [13] Yan-Qian, Y. et al. Theory of Limit Cycles, vol. 66 of Translations of Mathematical Monographs. Amer. Math. Soc., Providence, RI, 1986.

c CD-ROM Handbook of Differential Equations Academic Press 1997

17.

Natural Boundary Conditions for a PDE

17.

83

Natural Boundary Conditions for a PDE

Applicable to

Partial differential equations.

Yields A proper set of boundary conditions.

Idea Given a partial differential equation it is not always clear what the “correct” boundary conditions are. This is especially true for nonlinear partial differential equations. However, most partial differential equations that arise in mathematical physics have been obtained from a variational principle (see page 418). If we start with the variational principle, then “natural” boundary conditions will be generated while deriving the equation we started with. These boundary condition are, in a sense, the most appropriate boundary conditions for the original equation if there is no physical reason for imposing other conditions.

Procedure The variational principle that is most often used is δJ = 0, where δ represents a variation and J is a functional given by ZZ L(φ, φt , φx ) dt dx. J[φ] = R

Here L(·) is a linear or nonlinear functional and φ(x, t) is the unknown function to be determined. This variational principle states that the integral J[φ] should be stationary to small changes in φ. If we let h(x, t) be a continuously differentiable function, that is “small” in magnitude, then we can form ZZ n o J[φ + h] − J[φ] = Lφt ht + Lφxj hxj + Lφ dt dx + O(||h||2 ), R

where subscripts on L denote partial derivatives. The variational principle requires that δJ := J[φ + h] − J[φ] = 0, or that ZZ n o Lφt ht + Lφxj hxj + Lφ dt dx = 0. (17.1) R

If R is assumed to be a parallelpiped, then let Dt (Dxj ) denote the two parts of the boundary of R on which t (xj ) is constant. By integration by c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

parts, equation (17.1) can be written as  ZZ  ∂ ∂ − L φt − Lφ + Lφ h dt dx = 0, ∂t ∂xj xj

(17.2)

R

where we have assumed that Lφt = 0, Dt

Lφxj

= 0.

(17.3)

Dxj

Now h(x, t) was assumed to be arbitrary, so from equation (17.2) we conclude that ∂ ∂ Lφ + Lφ − Lφ = 0. ∂t t ∂xj xj

(17.4)

We conclude that if we can write a given partial differential equation in the form of equation (17.4) for some operator L(·), then equation (17.3) gives the “natural” boundary conditions.

Example Given the partial differential equation φtt − α2 ∇2 φ + β 2 φ = 0, where ∇2 φ =

PN j=1

(17.5)

φxj xj , we find that

L(φ, φt , φx ) =

N 1 2 1 2X 2 1 φt − α φxj − β 2 φ2 2 2 j=1 2

(17.6)

makes equations (17.4) and (17.5) identical. Therefore, the “natural” boundary conditions for equation (17.5) are, using equation (17.6) in (17.3), φt

= 0, Dt

φxj

= 0.

(17.7)

Dxj

Equation (17.7) states that the partial differential equation (17.5) requires both initial and boundary conditions. This was to be expected because equation (17.5) is a hyperbolic equation. For example, if N = 1 and R is the region [0, T ] × [0, ∞), then Dt = {t = 0} ∪ {t = T } and Dx1 = {x1 = 0} ∪ {x1 = ∞}. Hence, the natural boundary conditions for equation (17.5) require that {φt (0, x1 ), φt (T, x1 ), φx1 (t, 0), φx1 (t, ∞)} be specified. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Natural Boundary Conditions for a PDE

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Notes 1. Finding the operator L(·) or, equivalently, finding the variational principle δJ, is a non-trivial task in general. Also, very often one wants a vector variational principle that will encompass, simultaneously, several separate equations. 2. See the section on variational equations (on page 418) for more examples. 3. See also Kantorovich and Krylov [1, Chapter 4, pages 241–357] and Whitham [2].

References [1] Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis. Interscience Publishers, Inc., New York, 1958. [2] Whitham, G. B. Linear and Nonlinear Waves. Interscience Publishers, Inc., New York, 1974.

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18.

Definitions and Concepts

Normal Forms: Near-Identity Transformations

Applicable to

Systems of ordinary differential equations.

Yields A reformulation of the differential equations.

Idea Find a change of variables in the form of an infinite series, so that the original system of differential equations goes into a “normal” (or “simple” or “canonical”) form. The normal form is the simplest member of an equivalence class of differential equations, all exhibiting the same qualitative behavior. Normal forms are often useful for stability analyses.

Procedure

Start with the system x0 = f(x) such that (without loss of generality) x = 0 is a critical point. Expand this system to obtain x0 = Ax + H(x), where H(x) has strictly nonlinear functions (i.e., there are no linear or constant terms). If H(x) has nonlinear terms of at least degree n, then make a nearidentity transformation using polynomials of degree n with unknown coefficients. By appropriately choosing the unknown coefficients in the nearidentity transformation, the original differential equations, when written in the new variables, will have increased the degree of the nonlinear terms by one. We can summarize the procedure as follows: • We are given the system of ordinary differential equations x0 = f(x) = Ax + H(x), which we wish to analyze near the point x = 0. • We make the near-identity transformation from x to u via x = u + g(u), where g( ) is a strictly nonlinear function. • This change of variables produces the new equation u0 = [I + J]−1 f(u + g(u)) = Au + K(u),

(18.1)

∂g where I is the identity matrix and J = ∂u is the Jacobian of the transformation. • The function g( ) is chosen to eliminate the nonlinear terms in the equation for u that are of least order.

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This procedure can be iterated. If the critical point is “hyperbolic” (all eigenvalues have non-zero real parts), then the nonlinear terms can always be removed (i.e., one order at a time). Also, the topological nature does not change. See Guckenheimer and Holmes [7, Section 3.3].

Example 1 Suppose we have the system of equations dx = x + y2, dt dy = y + xy. dt  T Defining x = x y , this system has the form    2   dx 1 0 y 1 0 = x+ = x + H(x), 0 1 xy 0 1 dt

(18.2)

where H(x) has quadratic nonlinearities. We now choose to make the near-identity change of variables (of second order) x = u + a02 u2 + a11 uv + a20 v 2 , y = v + b02 u2 + b11 uv + b20 v 2 ,

(18.3)

where u and v are functions of t. Combining equation (18.2) and equation (18.3) we find du = u + (1 − a02 )v 2 − a11 uv − a20 u2 + higher order terms, dt dv = v − b02 v 2 + (1 − b11 )uv − b20 u2 + higher order terms, (18.4) dt where “higher order terms” means terms that are of order O(u3 , u2 v, uv 2 , v 3 ). To eliminate the second order terms in equation (18.4), we take {a02 = 1, a11 = 0, a20 = 0, b02 = 0, b11 = 1, b20 = 0}. With these values, the transformation in equation (18.3) becomes x = u + u2 , y = v + uv so that the original differential equations in (18.2) becomes du = u + higher order terms, dt dv = v + higher order terms. dt This new system now has cubic nonlinearities. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Example 2 The system of ordinary differential equations for x(t) and y(t): x0 = y + F (x, y),

(18.5)

y 0 = G(x, y), where F ( ) and G( ) are strictly nonlinear, has the normal form θ 0 = 1 + D1 r 2 + D2 r 4 + D3 r 6 + . . . , r0 =

B1 r3 + B2 r5 + B3 r7 + . . . ,

where u = r cos θ, v = r sin θ, and {u, v} are related, via a near-identity transformation, to {x, y}. In this example, the linear equations are not sufficient to determine the local behavior. Knowledge of B1 is needed to determine stability (unless it is zero, in which case B2 is needed, etc.). For example, if equation (18.5) has the form x2 y2 x3 x2 y + Fxy xy + Fyy + Fxxx + Fxxy 2 2 6 2 xy 2 y3 + Fyyy + ..., + Fxyy 2 6 x2 y2 x3 x2 y + Gxy xy + Fyy + Gxxx + Gxxy y 0 = Gxx 2 2 6 2 xy 2 y3 + Gyyy + ..., + Gxyy 2 6

x0 = y + Fxx

then we find (see Takens [12] for details) 16B1 = Gyyy + Gxxy + Fxyy + Fxxx + Fyy Gyy − Fxx Gxx − Gxx Gxy − Gyy Gxy + Fxx Fxy + Fxy Fyy .

Example 3 The system of ordinary differential equations for x(t) and y(t): x0 = −y + F (x, y), y0 =

(18.6)

x + G(x, y),

where F ( ) and G( ) are strictly nonlinear, has the normal form u0 = v +

∞ X n=2

bn u n ,

v0 =

∞ X

an u n ,

(18.7)

n=2

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Normal Forms: Near-Identity Transformations

89

where {u, v} are related, via a near-identity transformation, to {x, y}. For example, if equation (18.6) has the form x2 y2 x3 x2 y + Fxy xy + Fyy + Fxxx + Fxxy 2 2 6 2 xy 2 y3 + Fyyy + ..., + Fxyy 2 6 x2 y2 x3 x2 y + Gxy xy + Fyy + Gxxx + Gxxy x + Gxx 2 2 6 2 xy 2 y3 + Gyyy + ..., + Gxyy 2 6

x0 = −y + Fxx

y0 =

then we find that 1 1 (Gxy + Fxx ) u2 + (Gxy Gyy − Fxx Gyy 2 12 + 2Fxy Gxy + 2Gxxy − Fyy Gxx − Gxx + 2Fxxx ) u3 + . . . , (18.8) 1 1 v 0 = Gxx u2 + (3Fxy Gxx + Gxxx − Fxx Gxy ) u3 + . . . , 2 6

u0 = v +

where C is an arbitrary constant. See Takens [12] for details. Another normal form for equation (18.6) is given by U 0 = V, ∞ ∞ X X an U n + nbn U n−1 , U0 = n=2

n=2

where {U, V } are related, via a near-identity transformation, to {x, y}. See Guckenheimer and Holmes [7] for details.

Notes 1. If a2 6= 0, then the flow of the system in equation (18.7) is topologically equivalent to the flow of the system {u0 = v, v 0 = a2 u2 }, which can be integrated in terms of elliptic integrals. If a2 = 0, then other conclusions are possible; see Rand and Keith [11] for details. 2. To avoid computing the matrix inverse in equation (18.1), it is sufficient to expand (I + J)−1 into I − J + J 2 − · · · + (−J)n−1 if only the nonlinear terms of order n are to be removed. 3. The concept of normal forms does not require that the transformations used be near-identity ones, but they are the ones most often used in practice. 4. The computations needed for this technique quickly become unmanageable unless a computer algebra system is used. Macsyma programs for performing the necessary computations are given in Chow et al. [3] and in Rand and Keith [10]. 5. Abraham and Marsden [1, page 489] have the theorem c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

Consider the system L = K− P described by the Lagrangian P V where K = 12 i,j mij q˙i q˙j and V = 12 i,j cij qi qj and the matrices mij and cij are symmetric (this is no loss of generality) and mij is positive definite. Then there is a linear change of P P ˙ coordinates Qi = j aij qj and Qi = j aij q˙j such that the ¯ ¯ ¯ ¯ Lagrangian in the new P Pcoordinates is L = K − V where K = 1 1 2 ¯ ˙ i mi (Qi ) , V = 2 i ci Qi Qi , and mi > 0. 2 The new coordinates {Q1 , . . . , Qn , Q˙ 1 , . . . , Q˙ n } are called normal modes ¨ i + λ2 Qi = 0 (for i = 1, . . . , n) and Lagrange’s equations become Q i 2 where λi = −ci /mi .

References [1] Abraham, R., and Marsden, J. E. Foundations of Mechanics: A Mathematical Exposition of Classical Mechanics. Addison–Wesley Publishing Co., Reading, MA, 1994. [2] Ashkenazi, M., and Chow, S.-N. Normal forms near critical points for differential equations and maps. IEEE Trans. Circ. & Syst. 35, 7 (July 1988), 850–862. [3] Chow, S.-N., Byron, B., and Wang, D. Computation of normal forms. J. Comput. Appl. Math. 29, 2 (1990), 129–143. [4] Chua, L. O., and Kokubu, H. Normal forms for nonlinear vector fields— Part I: Theory and algorithm. IEEE Trans. Circ. & Syst. 35, 7 (July 1988), 863–880. [5] Chua, L. O., and Oka, H. Normal forms for constrained nonlinear differential equations—Part I: Theory. IEEE Trans. Circ. & Syst. 35, 7 (July 1988), 881–901. [6] Freire, E., Gamero, E., and Ponce, E. An algorithm for symbolic computation of Hopf bifurcation. In Computers and Mathematics, E. Kaltofen and S. M. Watt, Eds. Springer–Verlag, New York, 1990, pp. 109–118. [7] Guckenheimer, J., and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer–Verlag, New York, 1983. [8] Nayfeh, A. H. Method of Normal Forms. John Wiley & Sons, New York, 1993. [9] Rand, R. H., and Armbruster, D. Perturbation Methods, Bifurcation Theory and Computer Algebra. No. 65 in Applied Mathematical Sciences. Springer–Verlag, New York, 1987. [10] Rand, R. H., and Keith, W. L. Normal forms and center manifolds calculations on MACSYMA. In Applications of Computer Algebra, R. Pavelle, Ed. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1985. [11] Rand, R. H., and Keith, W. L. Determinacy of degenerate equilibria with linear part x’=y, y’=0 using MACSYMA. Appl. Math. and Comp. 21 (1987), 1–19. [12] Takens, F. Singularities of vector fields. Publ. Math. Inst. Hautes Etudes Sci. 43 (1974), 47–100. c CD-ROM Handbook of Differential Equations Academic Press 1997

19.

Random Differential Equations

19.

91

Random Differential Equations

Applicable to

Differential equations involving random terms.

Idea While randomness can appear in differential equations in many ways, most often it appears through “white noise” terms.

Procedure Suppose that x(t) is a random process that satisfies the stochastic differential equation dx(t) = a[x(t), t] dt + b[x(t), t] dw(t),

(19.1)

where w(t) is a standard Wiener process. The Wiener process is a Gaussian random process that has a mean given by its starting point, E [w(t)] =  w0 = w(t0 ), a variance of E (w(t) − w0 )2 = t − t0 , and a covariance of E [w(t)w(s)] = min(t, s). The sample paths of w(t) are continuous but not differentiable. If we define ∂b(x, t) 1 , α(x, t) = a(x, t) − b(x, t) 2 ∂x

(19.2)

then the solution to the stochastic differential equation, x(t), can be shown to satisfy (see Gardiner [5]) Z

Z

t

x(t) = x(t0 ) +

t

α[x(s), s] ds + t0

b[x(s), s] dw(s). S

t0

(19.3)

R where S represents the Stratonovich stochastic integral. Hence, an understanding of stochastic integration is required to understand the solutions to stochastic differential equations. If w(t) is a Wiener process and G(t, w(t)) is an arbitrary function, then Rt the stochastic integral I = t0 G(s, w(s)) dw(s) is defined as a limiting sum. Divide the interval [t0 , t] into n sub-intervals: t0 ≤ t1 ≤ · · · ≤ tn−1 ≤ tn = t, and choose points {τi } that lie in each sub-interval: ti−1 ≤ τi ≤ ti . The stochastic integral I is defined as the limit of partial sums, I = limn→∞ Sn , Pn with Sn = i=1 G(τi , w(τi ))[w(ti ) − w(ti−1 )]. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

Consider, for example, the special case of G(t) = w(t). Then the expectation of Sn is computed as " n # X w(τi )[w(ti ) − w(ti−1 )] E [Sn ] = E i=1

= =

n X i=1 n X

[min(τi , ti ) − min(τi , ti−1 )] (τi − ti−1 ).

i=1

PnIf we take τi = αti + (1 − α)ti−1 (where 0 < α < 1), then E [Sn ] = i=1 (ti − ti−1 )α = (t − t0 )α. Hence, the value of Sn depends on α. For consistency, some specific choice must be made for the points {τi }. R • For the Ito stochastic integral (indicated by I ), we choose τi = ti−1 (i.e., α = 0 in the above). That is ( n ) Z t X G(s, w(s)) dw(s) = ms-lim G(ti−1 , w(ti−1 ))[w(ti ) − w(ti−1 )] , I

n→∞

t0

i=1

(19.4) where ms-lim refers to the mean square limit. R • For the Stratonovich stochastic integral (indicated by S ), we choose τi = (ti + ti−1 )/2 (i.e., α = 1/2 in the above). That is (see Schuss [7]) Z S

t

G(w(s), x) dw(s) (19.5) ( n )   X  ti + ti−1 [w(ti ) − w(ti−1 )] . G ti−1 , w = ms-lim n→∞ 2 i=1

t0

in the evaluation of  seen R tintegrals can be R t The difference in these two 2 w(s) dw(s). We find that w(s) dw(s) = w (t) − w2 (t0 ) − (t − t0 ) /2 t0  2I t0 2  Rt while S t0 w(s) dw(s) = w (t) − w (t0 ) /2.

Notes 1. This book contains several sections for dealing with differential equations containing random terms: • To determine the transition probability density, see the discussion of the Fokker–Planck equation on page 303. • To obtain the moments without solving the complete problem, see pages 568 and 572. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Random Differential Equations

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• If the noise appearing in the differential equation is not “white noise,” the section on stochastic limit theorems might be useful (see page 629). • To numerically simulate the solutions of a stochastic differential equation, see the technique on page 775. 2. It can be shown that the Stratonovich integral has the usual properties of integrals, such as the fundamental theorem of integral calculus: Z t f 0 (w(s)) dw(s) = f (w(t)) − f (w(t0 )). S

t0

3. For arbitrary functions G, there is no connection between the Ito and Stratonovich integrals. However, when x(t) satisfies (19.1), then (see Gardiner [5, page 99]) Z Z t Z t 1 t ∂b[x(s), s] ds. b[x(s), s] dw(s) = b[x(s), s] dw(s)+ b[x(s), s] 2 t0 ∂x S t0 I t0 4. The Black–Scholes PDE for option pricing is obtained using stochastic differential equations (see Black and Scholes [1]). Let S represent the price of a share of stock, and assume S follows a geometric Brownian motion dS = µS dt+σS dω, where t is time, µ is a constant, and σ is the volatility constant. Let V (S, t) be the price of a derivative security whose payoff is only a function of S and t. Construct a portfolio consisting of V and ∆ shares of stock. The value P of this portfolio is P = V + ∆S. The differential of P is given by dP = dV + ∆ dS. Substituting for dV (using Ito’s lemma), and replacing dS by its assumed form results in     ∂V ∂V 1 2 2 ∂2V ∂V dP = + µS + σ S + σ∆S dω. + µ∆S + σS ∂t ∂S 2 ∂S 2 ∂S The random component of the portfolio increment can be removed by choosing ∆ = − ∂V ∂S . The concept of arbitrage says that dP = rP dt, where r is the (constant) risk-free bank interest rate. Combining the above results in the Black–Scholes PDE ∂V 1 ∂2V ∂V + rS + σ2 S 2 − rV = 0. ∂t ∂S 2 ∂S 2

References [1] Black, F., and Scholes, M. The pricing of options and corporate liabilities. J. Political Economy 81 (1973), 637–659. [2] Boyce, W. E. Random eigenvalue problems. In Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, Ed. Academic Press, New York, 1968, pp. 1–73. c CD-ROM Handbook of Differential Equations Academic Press 1997

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[3] Boyce, W. E. On a conjecture concerning the means of the eigenvalues of random Sturm–Liouville boundary value problems. Quart. Appl. Math. (1980), 241–245. [4] Day, W. B. Asymptotic expansions of eigenvalues and eigenfunctions of random boundary-value problems. Quart. Appl. Math. (July 1980), 169. [5] Gardiner, C. W. Handbook of Stochastic Methods. Springer–Verlag, New York, 1985. [6] Harlow, D. C., and Delph, T. J. The numerical solution of random intialvalue problems. Math. and Computers in Simulation 33 (1991), 243–258. [7] Schuss, Z. Theory and Applications of Stochastic Differential Equations. John Wiley & Sons, New York, 1980.

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20.

Self-Adjoint Eigenfunction Problems

20.

95

Self-Adjoint Eigenfunction Problems

Applicable to

Linear differential operators.

Yields Information that may be used to show completeness of a set of functions.

Procedure Many of the differential equations of mathematical physics are related to self-adjoint eigenfunction problems. As a special subcase, Sturm–Liouville equations are often self-adjoint eigenfunction problems. (Sturm–Liouville problems are discussed in more detail on page 103.) Let L[·] be the nth order linear operator defined by L[y] = pn (x)

dn y dn−1 y + p (x) + · · · + p0 (x)y, n−1 dxn dxn−1

where the {pi (x)} are complex valued and analytic and pn (x) 6= 0 on the interval x ∈ [a, b]. Define n boundary conditions by  n  X d(k−1) y d(k−1) y Bj [y] := Mjk (k−1) (a) + Njk (k−1) (b) = 0, j = 1, . . . , n, dx dx k=1

where the {Mjk , Njk } are given complex constants. The problem we consider is L[y] = λy,

B[y] = 0,

(20.1)

where B[y] = 0 is a shorthand notation for {Bj [y] = 0 | j = 1, . . . , n}. The system in equation (20.1) will always have the trivial solution, y(x) = 0. But, for certain values of λ, called eigenvalues, the system in equation (20.1) will have non-trivial solutions. Corresponding to the specific eigenvalue λn will be one or more eigenfunctions, that is, non-trivial solutions to (20.1) when λ = λn . We represent the complex conjugate of g by g¯. Define the inner prodRb g (t) dt and the norm of f (x) by uct of f (x) and g(x) by (f, g) = a f (t)¯ p ||f || := (f, f ). If (f, g) = 0, then f and g are said to be orthogonal. If {f1 , f2 , . . . , fn } are a set of functions with (fi , fj ) = 0 when i 6= j, then the {fi (x)} are an orthogonal family. The adjoint operator to L[·], called L∗ [·], is defined by L∗ [y] := (−1)n

(n−1) d(n) [¯ pn (x)y] [¯ pn−1 (x)y] n−1 d + (−1) + · · · + p¯0 y. dx(n) dx(n−1)

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Let u(x) be a solution to the system {L[u] = 0, B[u] = 0}, and let v(x) be a solution to the adjoint system {L∗ [v] = 0, B ∗ [v] = 0}, where {B ∗ [y] = 0} is a shorthand notation for {Bj∗ [y] = 0 | j = 1, . . . , n} and the Bi∗ [·] are, for the moment, unspecified. Using the definitions of u(x) and v(x), we can calculate vL[u] − uL∗ [v] =

d J(u, v), dx

(20.2)

where J(u, v) is called the bilinear concomitant and is defined by J(u, v) =

n X

X

 k

(−1)

m=1 j+k=m−1

 j  dk d v . (pm u) dxk dxj

(20.3)

Integrating equation (20.2) results in Z

b

a

    b (vL[u] − uL∗ [v]) dx = J(u, v) a = J u(b), v(b) − J u(a), v(a) . (20.4)

We now define the Bi∗ [·] to be those boundary conditions for which the right-hand side of equation (20.4) vanishes. If L = L∗ , then L is said to be formally self-adjoint. If L = L∗ and B = B ∗ , then L is said to be self-adjoint. Note that if L[·] is formally self-adjoint, then n = 2r and L[·] must be of the form     d dr dr u du b1 (x) + b0 (x)u. L[u] = r br (x) r + · · · + dx dx dx dx (20.5) As we now record, self-adjoint operators have some very useful properties. If L[·] is self-adjoint, then • The eigenvalues λn of equation (20.1) are real. • The eigenvalues are enumerable (with no cluster point). • The eigenfunctions yn (x) corresponding to distinct eigenvalues are orthogonal. • If f (x) is any analytic function that satisfies the boundary conditions in equation (20.1) (i.e., Bj [f ] = 0, for j = 1, . . . , n), then, on the ∞ X (f, yk ) yk (x). interval [a, b], we have the representation f (x) = (yk , yk ) k=0

That is, the {yk (x)} are complete. It is this last statement that is of particular importance in solving differential equations. The method suggested by this statement, the method of eigenfunction expansions, is described on page 268. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Self-Adjoint Eigenfunction Problems

97

Example 1 Suppose we have the linear differential operator     d2 d d2 y dy r1 (x) + r0 (x). L[y] = 2 r2 (x) 2 + dx dx dx dx

(20.6)

Because of the form of the operator, we know that L[·] will be formally selfadjoint (see equation (20.5)). For this operator, we can evaluate J(u, v) at the upper and lower limits (from equation (20.3)) to find i b h b J(u, v) a = v(r2 u00 )0 − v 0 r2 u00 + r2 v 00 u0 − u(r2 v 00 )0 + r1 (vu0 − uv 0 ) a . (20.7) To determine whether L[·] is self-adjoint or not, we need to specify B[y]. Because equation (20.6) is a fourth order operator, four boundary conditions are required. We will consider three separate cases: • Case 1 If B[y] is defined by B1 [y] = y(a), B2 [y] = y 00 (a), B3 [y] = y(b), B4 [y] = y 00 (b),

(20.8)

then J(u, v) can be evaluated and equation (20.7) can be simplified to yield b (20.9) r2 v 00 u0 + r1 vu0 a . If we choose B = B ∗ (i.e., Bi∗ [y] = Bi [y]), then the quantity in (20.9) is identically zero. Hence, L[·], as defined by equations (20.6) and (20.8) is self-adjoint. • Case 2 If B[y] is defined by B1 [y] = y(a), B2 [y] = y 0 (a), B3 [y] = y(b),

(20.10)

B4 [y] = y 0 (b), then J(u, v) can be evaluated and equation (20.7) can be simplified to yield b (20.11) v(r2 u00 )0 − v 0 r2 u00 a . Once again, if we choose B = B ∗ , then the quantity in (20.11) is identically zero. Hence, L[·], as defined by equations (20.6) and (20.10) is self-adjoint. c CD-ROM Handbook of Differential Equations Academic Press 1997

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• Case 3 If B[y] is defined by B1 [y] = y(a), B2 [y] = y 0 (a), B3 [y] = y 00 (a), B4 [y] = y 000 (a),

(20.12)

then J(u, v) can be evaluated and equation (20.7) can be simplified to yield v(r2 u00 )0 − v 0 r2 u00 + r2 v 00 u0 − u(r2 v 00 )0 + r1 (vu0 − uv 0 ) x=b . (20.13) If, in this case, we choose B = B ∗ , then the quantity in equation (20.13) does not vanish. If B = B ∗ , then no information has been given at the boundary x = b, and the quantity in (20.13) is indeterminate. Hence, L[·], as defined by equations (20.6) and (20.12), is not self-adjoint. An initial value problem can never be self-adjoint.

Example 2 The operator d L[y] = dx

  dy dy a2 (x) + a1 (x) + a0 (x), dx dx

with the boundary conditions B1 [y] = y(a), B2 [y] = y 0 (b), is self-adjoint. See the section on Sturm–Liouville theory (page 103).

Example 3 A third order linear ordinary differential equation is formally self-adjoint if it has the form      d d2 y dy dy d  d2 + P (x) Q(x)y + Q(x) = 0. P (x) + 2 2 dx dx dx dx dx dx (20.14) The general third order linear ordinary differential equation A(x)

d2 y dy d3 y + D(x) = 0, + B(x) + C(x) dx3 dx2 dx


0 will be formally self-adjoint if and only if B = 32 A0 and D = 12 C − 13 B 0 . The self-adjoint third order equation (20.14) has the first integral
P 2yy 00 − (y 0 )2 + P 0 yy 0 + Qy 2 = constant. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Example 4 The general fourth order linear ordinary differential equation A(x)y 0000 + B(x)y 000 + C(x)y 00 + D(x)y 0 + E(x)y = 0,
0 will be formally self-adjoint if and only if B = 2A0 and D = C − 12 B 0 .

Notes 1. Some of the conditions above can be relaxed, and the main results for self-adjoint operators will still be true. See, for instance, Coddington and Levinson [3, Chapter 7]. 2. For partial differential equations there are many results analogous to those mentioned above for ordinary differential equations. We enumerate some of them for the Helmholtz equation in two dimensions: For the equation ∇2 φ + λφ = 0, in a region R, with the boundary conditions aφ + b∇φ · n = 0, given on the entire boundary of R (here n represents the unit normal): • All the eigenvalues {λi } are real. • There are an infinite number of eigenvalues. There is an eigenvalue of least magnitude but no largest one. • The eigenfunctions {φi (x, y)} form a complete set:P Any analytic function can be represented in the form f (x, y) = i ai φi (x, y), for some set of constants {ai }. • Eigenfunctions belonging to different eigenvalues are orthogonal. RR φi φ¯j dx dy = 0, if λi 6= λj . That is R

• An eigenfunction φ is related to it’s eigenvalue λ by the Rayleigh quotient H RR − φ∇φ · n ds + |∇φ|2 dx dy R RR . λ= φ2 dx dy R

3. Many other partial differential equations have very similar properties. See Haberman [5, pages 214–219] for details. 4. Partial differential equations can also be self-adjoint. The elliptic equation auxx +cuyy +dux +euy +f u = g(x, y) is said to be essentially self-adjoint when Nx = My , where N := d −

ax , a

M := e −

cy . c

In this case, an integrating factor is given by eφ , where φx = N , φy = M . Multiplying the original equation by this factor puts the equation in self-adjoint form. For example, the equation uxx + uyy + x2 ux + y 2 uy + u = 0 c CD-ROM Handbook of Differential Equations Academic Press 1997

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has N = x2 , M = y 2 , which leads to φ = 13 x3 + y 3 . Multiplying the equation by eφ results in the self-adjoint form of the equation: i i h 3 3 h 3 3 3 3 e(x +y )/3 ux + e(x +y )/3 uy + e(x +y )/3 u = 0. x

y

5. See Birkhoff and Rota [1, Chapters 10–11], Butkov [2, Chapter 9, pages 332–404], Dunford and Schwartz [4], Ince [6, Chapters 9–11, pages 204–278], and Stakgold [7, Chapter 3].

References [1] Birkhoff, G., and Rota, G.-C. Ordinary Differential Equations. John Wiley & Sons, New York, 1978. [2] Butkov, E. Mathematical Physics. Addison–Wesley Publishing Co., Reading, MA, 1968. [3] Coddington, E. A., and Levinson, N. Theory of Ordinary Differential Equations. McGraw–Hill Book Company, New York, 1955. [4] Dunford, N., and Schwartz, J. Linear Operators, Part II: Spectral Theory. John Wiley & Sons, New York, 1958. [5] Haberman, R. Elementary Applied Partial Differential Equations. Prentice– Hall, Inc., Englewood Cliffs, NJ, 1968. [6] Ince, E. L. Ordinary Differential Equations. Dover Publications, Inc., New York, 1964. [7] Stakgold, I. Green’s Functions and Boundary Value Problems. John Wiley & Sons, New York, 1979.

c CD-ROM Handbook of Differential Equations Academic Press 1997

21.

Stability Theorems

21.

101

Stability Theorems

Applicable to

Differential equations of all types.

Yields Knowledge of whether or not there are stable solutions.

Idea There are theorems available for most cases of interest.

Procedure There are many theorems that can be used to determine whether the solutions to a differential equation are stable. For example, useful simple theorems include Theorem Consider the equation y0 = Ay + f(t, y), where A is a real constant matrix whose eigenvalues all have negative real parts. Let f be real, continuous for small |y| and t ≥ 0, and f(t, y) = o(|y|) as |y| → 0, uniformly for t ≥ 0. Then the identically zero solution is asymptotically stable. Theorem If 1. Every solution of y0 = Ay approaches zero as t → ∞, 2. ||f (z)||/||z|| → 0 as z → 0, 3. ||f (z1 ) − f (z2 )|| ≤ c1 ||z1 − z2 || for ||z1 || and ||z2 || less than c2 where c1 → 0 as c2 → 0, then z = 0 is a stable solution of y0 = Ay + f (y).

Example Consider the equation y 0 = −2y + f (t). Using the second theorem the solution y = 0 is stable for f (y) = y n when n > 1.

Notes 1. Stability is required if a differential equation is to be well posed (see page 115). 2. Floquet theory and Lyapunov functions are two techniques that can determine whether an equation has stable or unstable solutions (see pages 523 and 551). 3. Note that solutions to the equation y0 = A(t)y can be increasing even if all the eigenvalues of A(t) have negative real parts for any fixed value of#t. For example, consider the matrix A(t) = " 1 1 − 4(1+t) −1 ± 2i (1+t)2 , . This matrix has the eigenvalues λ1,2 = 1 − 14 − 4(1+t) 4(1 + t) c CD-ROM Handbook of Differential Equations Academic Press 1997

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yet the general solution to y0 = A(t)y is given by     (1 + t)−3/4 log(1 + t)
(1 + t)−3/4 , + β y(t) = α − 21 (1 + t)1/4 (1 + t)1/4 1 − 12 log(1 + t) where α and β are arbitrary constants. 4. There are many different technical definitions of stability. For the equation y0 = f(t, y),

(21.1)

defined when t ≥ t0 , the solution is said to be • Stable if for each  > 0 there is a corresponding δ = δ() > 0 b (t) of equation (21.1) that satisfies the such that any solution y inequality |b y(t0 ) − y(t0 )| <  exists and satisfies the inequality |b y(t) − y(t)| < δ for all t ≥ t0 . A solution that is not stable is said to be unstable. • Asymptotically stable if, in addition to the above stability requirements, |b y(t) − y(t)| → 0 as t → ∞, whenever |b y(t0 ) − y(t0 )| is sufficiently small. • Uniformly stable if for each  > 0 there is a corresponding δ = b(t) of equation (21.1) that δ() > 0 such that any solution y satisfies the inequality |b y(t0 ) − y(t0 )| < δ for some t1 ≥ t0 exists and satisfies the inequality |b y(t) − y(t)| <  for all t ≥ t1 . • Uniformly asymptotically stable if, in addition to the requirements for asymptotic stability, there is a δ0 > 0, and for each  > 0 a corresponding T = T () > 0 such that if |b y(t1 ) − y(t1 )| < δ0 for some t1 ≥ t0 , then |b y(t) − y(t)| <  for all t ≥ t1 + T . • Strongly stable if for each  > 0 there is a corresponding δ = b(t) of equation (21.1) that δ() > 0 such that any solution y satisfies the inequality |b y(t0 ) − y(t0 )| < δ for some t1 ≥ t0 exists and satisfies the inequality |b y(t) − y(t)| <  for all t ≥ t0 .

References [1] Bellman, R. Stability Theory of Differential Equations. McGraw–Hill Book Company, New York, 1953. [2] Boyce, W. E., and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, fourth ed. John Wiley & Sons, New York, 1986. [3] Coddington, E. A., and Levinson, N. Theory of Ordinary Differential Equations. McGraw–Hill Book Company, New York, 1955.

c CD-ROM Handbook of Differential Equations Academic Press 1997

22.

Sturm–Liouville Theory

22.

103

Sturm–Liouville Theory

Applicable to

Second order linear ordinary differential operators.

Yields Information about whether an operator is self-adjoint.

Procedure Many of the differential equations of mathematical physics are Sturm– Liouville equations. Sturm–Liouville equations arise naturally, for instance, when separation of variables (see page 487) is applied to the wave equation, the potential equation, or the diffusion equation. The Sturm–Liouville operator, L, is defined by     d d 1 − p(x) + q(x) , (22.1) L := s(x) dx dx where p, p0 , q, and s are real and continuous and s(x) > 0 and p(x) > 0 on the interval (a, b). The Sturm–Liouville equation is defined by L[y(x)] = −λy(x),

(22.2)

or, equivalently, −

  dy d p(x) + q(x)y + λs(x)y = 0, dx dx

(22.3)

for x ∈ [a, b]. The parameter λ is an eigenvalue of the equation. Given a specific set of boundary conditions, there may be specific values of λ for which equation (22.2) has a non-trivial solution. For different types of boundary conditions, different types of behavior are possible. Many facts are known about Sturm–Liouville systems: • L, as defined by equation (22.1), is formally self-adjoint (see page R 95), with the inner product, (f, g)s := s(x)f (x)¯ g (x) dx. • L is self-adjoint (see page 95) when – The boundary conditions are unmixed (or separated). That is, they are of the form α1 y(a) + β1 y 0 (a) = 0, α2 y(b) + β2 y 0 (b) = 0.

(22.4)

– The boundary conditions are periodic. That is, they are of the form y(a) = y(b), y 0 (a) = y 0 (b). c CD-ROM Handbook of Differential Equations Academic Press 1997

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• When the boundary conditions are given as in equation (22.4), and, in addition, p(x) > 0, q(x) > 0, α1 /β1 > 0, α2 /β2 > 0, then – L is a positive definite operator (i.e., (Lu, u) > 0, for all u 6= 0). – The eigenvalues are simple (i.e., each eigenvalue has a single eigenfunction associated with it). • When the operator L is not self-adjoint then ¯ is an eigenvalue of L∗ , – If λ is a complex eigenvalue of L, then λ the adjoint of L. – Eigenfunctions of L are orthogonal to those of L∗ . If the interval [a, b] is finite and p(x) and s(x) are positive at the endpoints, then the problem is said to be regular. Otherwise, it is said to be singular. For singular Sturm–Liouville problems, problems are subdivided into two cases, the limit-circle case and limit-point case. Consider equation (22.2) when one of the endpoints is regular and the other singular. Define the s-norm of a function u(x) by Z

b

||u||s = (u, u)s =

s(x)|u(x)|2 dx. a

If, for any particular complex number λ, the solution to equation (22.2) satisfies • ||y||s < ∞, then L is said to be of the limit-circle type at infinity. In this case, all solutions of equation (22.2) will satisfy ||y||s < ∞, for any value of λ. • ||y||s = ∞, then L is said to be of the limit-point type at infinity. If both endpoints are singular, we introduce an intermediate point l, a < l < b and then classify L as being of the limit-point type or the limit-circle type at each endpoint according to the behavior of solutions in a < x < l and in l < x < b (the classification is independent of the choice of l). For a given real λ, the problem in equation (22.2) is • Oscillatory at x = a if and only if every solution has infinitely many zeros clustering at a. • Nonoscillatory at x = a if and only if no solution has infinitely many zeros clustering at a. The classification is mutually exclusive for a fixed λ but can vary with λ. If L is in the limit-point case at infinity, then there is the following completeness theorem: R∞ R∞ Theorem If g(λ) = 0 f (x)Ψ(x, λ) dx, then f (x) = −∞ g(λ) Ψ(x, λ) dρ(λ) for a (computable) density function ρ(λ). c CD-ROM Handbook of Differential Equations Academic Press 1997

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A completeness theorem is required for a proof that a separation of variables calculation (see page 487) has been done correctly. The following theorem and corollaries may help decide the type of the operator L: Theorem Let M be a positive differentiable function, and let k1 and k2 be two positive constants such that for large x, q(x) ≥ −k1 M (x), Z ∞ (p(t)M (t))−1/2 dt = ∞, x

|p1/2 (x)M 0 (x)M −3/2 (x)| < k2 , then L is in the limit-point case at infinity. If q(x) ≥ −k, where k is a positive constant, and RCorollary ∞ −1/2 p (t) dt = ∞ (where n is any finite number), then L is in the n limit-point case at infinity. Corollary If p(x) = 1 for 0 < x < ∞ and q(x) ≥ −kx2 for some positive constant k, then L is in the limit-point case at infinity.

Example 1 The differential equation and boundary conditions −(xy 0 )0 = λxy, u(1) = 0, u(2) = 0, correspond to the Sturm–Liouville operator in equation (22.1) with p(x) = x, q(x) = 0, and s(x) = x. This is a regular Sturm–Liouville problem on the interval [1, 2]. The eigenvalues and eigenfunctions are readily computed (see Stakgold [6, page 423]. If we define λn = rn2 , then the rn are determined from N0 (rn ) J0 (rn ) = , J0 (2rn ) N0 (2rn ) and the corresponding eigenfunction is given by rn πJ0 (2rn ) yn (x) = √ p [J0 (rn )N0 (rn x) − J0 (rn x)N0 (rn )]. 2 J0 (rn )2 − J0 (2rn )2

Example 2 The differential equation with boundary conditions −(x2 y 0 )0 − λu = 0, u(1) = 0, u(e) = 0, c CD-ROM Handbook of Differential Equations Academic Press 1997

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for x ∈ [1, e] is a regular Sturm–Liouville problem with unmixed boundary conditions, so the eigenfunctions are complete. In this case we find 1 λn = n2 π 2 + , yn = x−1/2 sin(nπ log x). 2

22.1

Classification of Sturm–Liouville Problems

Pruess et al. [5] have devised a classification scheme and taxonomy for Sturm–Liouville problems on the interval (a, b). They define: Category 1: Problem (22.2) is nonoscillatory at x = a and x = b. The spectrum is simple, purely discrete, and bounded below. Category 2: Problem (22.2) is nonoscillatory at one endpoint. At the other endpoint, it is nonoscillatory for λ ∈ (−∞, t0 ) and oscillatory for λ ∈ (t0 , ∞). The spectrum is simple and bounded below. The point spectrum (if any) is in (−∞, t0 ) whereas (t0 , ∞) is the continuous spectrum. Category 3: Problem (22.2) is nonoscillatory at one endpoint. At the other endpoint it is limit-circle and oscillatory. The spectrum is simple, unbounded both above and below, and purely discrete. Category 4: Problem (22.2) is nonoscillatory at one endpoint. At the other endpoint, it is limit-point and oscillatory. The spectrum is simple and purely continuous; the continuous spectrum is the entire real line. Category 5: Problem (22.2) is limit-circle and oscillatory at x = a. It is limit-point and oscillatory at x = b. The spectrum is simple, unbounded both above and below, and purely discrete. Category 6: Problem (22.2) is limit-point and oscillatory at x = a. It is limit-point and oscillatory at x = b. The nature of the spectrum is unknown; a continuous spectrum is likely. Category 7: Problem (22.2) is limit-point and oscillatory at one endpoint (x = a or x = b). At the other endpoint, it is limit-circle and oscillatory. The spectrum is simple and purely continuous; the continuous spectrum is the entire real line. Category 8: Problem (22.2) is limit-circle and oscillatory at one endpoint (x = a or x = b). At the other endpoint, it is nonoscillatory for λ ∈ (−∞, t0 ) and oscillatory for λ ∈ (t0 , ∞). The spectrum is simple; the point spectrum (if any) is unbounded below but bounded above by t0 . The continuous spectrum is in (t0 , ∞). c CD-ROM Handbook of Differential Equations Academic Press 1997

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Category 9: Problem (22.2) is limit-point and oscillatory at one endpoint (x = a or x = b). At the other endpoint, it is nonoscillatory for λ ∈ (−∞, t0 ) and oscillatory for λ ∈ (t0 , ∞). The spectrum may be nonsimple. Category 10: At x = a problem (22.2) is nonoscillatory for λ ∈ (−∞, t0 ) and oscillatory for λ ∈ (t0 , ∞). At x = b, it is nonoscillatory for λ ∈ (−∞, t1 ) and oscillatory for λ ∈ (t1 , ∞). The spectrum may be nonsimple. The point spectrum (if any) is in the interval (−∞, min(t0 , t1 )) and is bounded below. The continuous spectrum is in (min(t0 , t1 ), ∞).

Notes 1. For transformations of equation (22.3), see page 157. 2. The regular Sturm–Liouville equation, written in the form d2 z − r(t)z + λz = 0, dt2 with the boundary conditions z(0) = z(L) = 0, has the asymptotic eigenvalues and eigenfunctions r    nπ  2 1 zn (t) = sin t +O , L L n n2 π 2 + O(1) λn = L2 as n → ∞. (See the Pr¨ ufer method on page 150.) 3. For the Sturm–Liouville equation L[y] = −(py 0 )0 + qy − λwy = 0 on [a, ∞], define θ and φ to be solutions satisfying {θ(a) = 0, pθ0 x=a = 1} and {φ(a) = −1, pφ0 x=a = 0}. The Titchmarsh–Weyl function m(λ) is defined to be the functions {m± }, defined on the upper and R∞ 2 lower half planes, such that a |θ(x, λ) + m± (λ)φ(x, λ)| dx < ∞ for all strictly complex values of λ. 4. See also Birkhoff and Rota [1, Chapters 10–11], Coddington and Levinson [2, Chapters 7–12], Levitan and Sargsjan [4, Chapter 6, pages 139–182 and Chapter 12, pages 324–340], Stakgold [6, Chapter 7, pages 411–466], and Zauderer [7, pages 136–159].

References [1] Birkhoff, G., and Rota, G.-C. Ordinary Differential Equations. John Wiley & Sons, New York, 1978. [2] Coddington, E. A., and Levinson, N. Theory of Ordinary Differential Equations. McGraw–Hill Book Company, New York, 1955. [3] Dunford, N., and Schwartz, J. Linear Operators, Part II: Spectral Theory. John Wiley & Sons, New York, 1958. c CD-ROM Handbook of Differential Equations Academic Press 1997

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[4] Levitan, B. M., and Sargsjan, I. S. Sturm–Liouville and Dirac Operators. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. [5] Pruess, S., Fulton, C. T., and Xie, Y. The automatic classification of Sturm–Liouville problems. Appl. Math. and Comp.. (submitted for publication). [6] Stakgold, I. Green’s Functions and Boundary Value Problems. John Wiley & Sons, New York, 1979. [7] Zauderer, E. Partial Differential Equations of Applied Mathematics. John Wiley & Sons, New York, 1983.

c CD-ROM Handbook of Differential Equations Academic Press 1997

23.

Variational Equations

23.

109

Variational Equations

Applicable to

Differential equations that arise from variational

principles.

Yields A variational principle.

Procedure Most differential equations that arise in mathematical physics have been obtained from a variational principle. The variational principle that is most often used is δJ = 0, where δ represent a variation and J is a functional given by ZZ J[u] = L(x, ∂xj )u(x) dx. (23.1) R

Here, L(·) is a linear or nonlinear function of its arguments, and u(x) is the unknown function to be determined. This variational principle states that the integral J[u] should be stationary to small changes in u(x). If we let h(x) be a “small,” continuously differentiable function, then we can form ZZ  J[u + h] − J[u] = L(x, ∂xj )(u(x) + h(x)) − L(x, ∂xj )u(x) dx. (23.2) R By integration by parts, equation (23.2) can often be written as ZZ N (x, ∂xj )u(x) dx + O(||h||2 ), J[u + h] − J[u] = R

plus some boundary terms (see page 83). The variational principle requires that δJ := J[u + h] − J[u] vanishes to leading order, or that N (x, ∂xj )u(x) = 0.

(23.3)

Equation (23.3) is called the first variation of equation (23.1) or the Euler– Lagrange equation corresponding to equation (23.1). (This is sometimes called the Euler equation.) A functional in the form of equation (23.1) determines an Euler–Lagrange equation. Conversely, given an Euler–Lagrange equation, a corresponding functional can sometimes be obtained. Many approximate and numerical techniques utilize the functional associated with a given system of Euler–Lagrange equations. See, for example, the Rayleigh–Ritz method (page 638) and the finite element method (page 734). c CD-ROM Handbook of Differential Equations Academic Press 1997

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The following collection of examples assumes that the dependent variable in the given differential equation has natural boundary conditions (see page 83). If the dependent variable did not have these specific boundary conditions, then the boundary terms that were discarded in going from equation (23.2) to equation (23.3) would have to be satisfied in addition to the Euler–Lagrange equation.

Example 1 The Euler–Lagrange equation for the functional Z   F x, y, y 0 , . . . , y (n) dx, J[y] =

(23.4)

R

where y = y(x) is       n d ∂F d2 ∂F ∂F ∂F n d − + 2 − · · · + (−1) = 0. ∂y dx ∂y 0 dx ∂y 00 dxn ∂y (n) (23.5) For this equation the natural boundary conditions are given by y(x0 ) = y0 ,

y 0 (x0 ) = y00 ,

y(x1 ) = y1 ,

y 0 (x1 ) = y10 ,

... , ... ,

(n−1)

y (n−1) (x0 ) = y0 y (n−1) (x1 ) =

,

(n−1) y1 .

Example 2 The Euler–Lagrange equation for the functional ZZ F (x, y, u, ux , uy , uxx , uxy , uyy ) dx dy, J[u] =

(23.6)

R

where u = u(x, y) is ∂ ∂F − ∂u ∂x



  ∂ ∂F − + ∂y ∂uy   ∂2 ∂F + + ∂x∂y ∂uxy

∂F ∂ux



∂2 ∂x2 ∂2 ∂y 2

 

∂F ∂uxx ∂F ∂uyy

  = 0.

(23.7)

Example 3 The Euler–Lagrange equation for the functional #  2 Z Z "  2 ∂u ∂u a +b + cu2 + 2f u dx dy, J[u] = ∂x ∂y R

is ∂ ∂x

    ∂u ∂ ∂u a + b − cu = f. ∂x ∂y ∂y

(23.8)

(23.9)

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Variational Equations

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Example 4 For the 2mth order ordinary differential equation (in formally selfadjoint form)   m k X dk u k d pk (x) k = f (x), (−1) dxk dx k=0 (23.10) u(a) = u0 (a) = · · · = u(m−1) (a) = 0, u(b) = u0 (b) = · · · = u(m−1) (b) = 0, a corresponding functional is !  k 2 Z b X m d u pk (x) − 2f (x)u(x) dx. J[u] = dxk a k=0

(23.11)

Example 5 Consider the system of n second order ordinary differential equations for the unknowns {uk (x) | k = 1, . . . , n}    n  X duk d pjk (x) + qjk (x)uk = fj (x), − dx dx (23.12) k=1 uj (a) = uj (b) = 0, for j = 1, 2, . . . , n. If pjk = pkj , qjk = qkj , if the matrix {pjk } is bounded and positive definite, and if the matrix {qjk } is bounded and non-negative definite, then a functional corresponding to equation (23.12) is    X Z b X n  n du du j k  + qjk (x)uj uk − pjk (x) fj (x)uj (x) dx. J[u] = dx dx a j=1 j,k=1 (23.13)

Example 6 If Aij (x) is a symmetric and positive definite matrix, so that the partial differential equation for u(x) = u(x1 , . . . , xm )   m X ∂ ∂u − Aij + C(x)u = f (x), (23.14) ∂xi ∂xj i,j=1 is elliptic in Ω, C(x) > 0, and there are Dirichlet boundary conditions u ∂Ω = 0, (23.15) then a corresponding functional is   Z m X ∂u ∂u  J[u] = Aij + Cu2 − 2f u dx, ∂x ∂x i j Ω i,j=1

(23.16)

where (23.16) is to be minimized over those functions that satisfy equation (23.15). c CD-ROM Handbook of Differential Equations Academic Press 1997

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Example 7 If Aij (x) is a symmetric and positive definite matrix, so that the partial differential equation for u(x) = u(x1 , . . . , xm ),   m X ∂ ∂u Aij + C(x)u = f (x), (23.17) − ∂xj ∂xj i,j=1 is elliptic in Ω, C(x) > 0, and there are the boundary conditions   m X ∂u  Aij cos(ν, xi ) + σu = 0, ∂x j i,j=1

(23.18)

∂Ω

where ν is normal to ∂Ω and σ is a positive function on ∂Ω, then a corresponding functional is   Z Z m X ∂u ∂u  Aij + Cu2 − 2f u dx + σu2 dS, J[u] = ∂x ∂x i j Ω ∂Ω i,j=1 (23.19) where (23.19) is to be minimized over those functions for which equation (23.18) is satisfied.

Notes 1. Note that two different functionals Rcan yield the R same set0 of Euler– Lagrange equations. For example, δ J dx = δ (J +y +xy ) dx. The R reason that δ (yR + xy 0 ) dx = 0 Ris because the integrand is an exact differential (i.e., (y +xy 0 ) dx = d(xy)). Hence, this integral is path independent; its value is determined by the boundary RR conditions. The Euler–Lagrange equations for the two functionals uxx uyy dx dy and RR 2 (uxy ) dx dy are also the same. 2. If a differential equation can be derived from a variational principle, then admittance of a Lie group is a necessary condition to find conservation laws by Noether’s theorem. 3. Even if the boundary conditions given with a differential equation are not natural, a variational principle may sometimes be found. Consider Z x2 F (x, u, u0 ) dx − g1 (x, u) x=x + g2 (x, u) x=x , J[u] = 1

x1

2

where g1 (x, u) and g2 (x, u) are unspecified functions. The necessary conditions for u to minimize J[u] are (see Mitchell and Wait [5]). d ∂F ∂F − = 0, ∂u dx ∂u0

∂g1 ∂F + = 0, ∂u0 ∂u x=x1

∂F ∂g2 + = 0. ∂u0 ∂u x=x2

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If g1 and g2 are identically zero, then we recover the natural boundary conditions. However, we may choose g1 and g2 to suit other boundary conditions. For example, the problem u00 + f (x) = 0, u0 + αu x=x = 0, 1

u0 + βu x=x = 0 2

corresponds to the functional  Z x2  βu2 αu2 1 0 2 (u ) − f (x)u dx + J[u] = − . 2 2 x=x2 2 x=x1 x1 4. This technique can be used in higher dimensions. For example, consider the functional ZZ F (x, y, u, ux, uy , uxx , uxy , uyy ) dx dy J[u] = R

Z G(x, y, u, uσ , uσσ , un ) dσ,

+ ∂R

where ∂/∂σ and ∂/∂n are partial differential operators in the directions of the tangent and normal to the curve ∂R. Necessary conditions for J[u] to have a minimum are the Euler–Lagrange equations (given in equation (23.7)) together with the boundary conditions:    ∂ ∂F ∂ ∂F ∂F ∂F yσ − xσ − − ∂ux ∂x ∂uxx ∂uy ∂y ∂uyy     
∂F 1 ∂ ∂F ∂F ∂ 2 2 xσ yσ + − − x − yσ ∂σ ∂uxx ∂uyy 2 ∂σ ∂uxy σ      ∂ ∂F 1 ∂ ∂F xσ − yσ + 2 ∂x ∂uxy ∂y ∂uxy (23.20) ∂ ∂G ∂ 2 ∂G + = 0, + Gu − ∂σ ∂uσ ∂σ 2 ∂uσσ ∂F 2 ∂F 2 ∂F ∂G + y + x + xσ yσ = 0, ∂un ∂uxx σ ∂uyy σ ∂uxy 

dy dx and yσ = dσ . See Mitchell and Wait [5] for details. where xσ = dσ 5. Mathematica has the package VariationalMethods which can determine the Euler equations for a general integrand. 6. See also Butkov [1, pages 573–588], Collatz, [2, pages 540–541], Farlow, [3, pages 362–369], and Kantorovich and Krylov [4, Chapter 4, pages 241–357].

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References [1] Butkov, E. Mathematical Physics. Addison–Wesley Publishing Co., Reading, MA, 1968. [2] Collatz, L. The Numerical Treatment of Differential Equations. Springer– Verlag, New York, 1966. [3] Farlow, S. J. Partial Differential Equations for Scientists and Engineers. John Wiley & Sons, New York, 1982. [4] Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis. Interscience Publishers, Inc., New York, 1958. [5] Mitchell, A. R., and Wait, R. The Finite Element Method in Differential Equations. John Wiley & Sons, New York, 1977. [6] Yourgrau, W., and Mandelstam, S. Variational Principles in Dynamics and Quantum Theory. Dover Publications, Inc., New York, 1979.

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Well Posed Differential Equations

24.

115

Well Posed Differential Equations

Applicable to

Ordinary and partial differential equations.

Yields Knowledge of whether the equation is intrinsically well posed.

Idea Before an attempt is made to determine or approximate the solution of a differential equation, it should be checked to determine if the differential equation problem is intrinsically well posed.

Procedure A well posed differential equation is one in which • The solution exists. • The solution is unique. • The solution is stable (i.e., the solution depends continuously on the boundary conditions and initial conditions). If the differential equation is not well posed, it is called an ill posed or improperly posed problem. For such problems, there may not be a solution, there may be more than one solution, or whatever solution is determined (by an approximate scheme) may be unrelated to the actual solution. For partial differential equations, the third condition (concerning stability) is generally the easiest to check.

Example Consider the initial value problem for the unknown function u(x, t), utt = uxxxx, u(x, 0) = g(x).

(24.1)

We will show that the solution to this problem is not stable. Suppose that equation (24.1) has a solution, say u0 (x, t). Assume that  is a fixed number, much smaller than one in magnitude, and define u1 (x, t) by u1 (x, t) = u0 (x, t) + eikx eσt , where k and σ are also constants. At t = 0, u1 (x, 0) differs from g(x) by a quantity that has magnitude , an arbitrarily small amount. However, using u1 (x, t) in equation (24.1), we determine that u1 (x, t) will satisfy the equation if σ = ±k 2 . Therefore, at any fixed value of t, say t = T , there exists a solution u0 (x, T ) and an approximation to the c CD-ROM Handbook of Differential Equations Academic Press 1997

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2

solution u1 (x, T ) = u0 (x, T ) + eikx ek T . The approximation satisfies the same differential equation that the true solution satisfies. But because k is arbitrary, the approximate solution can be arbitrarily larger than the true solution by making k arbitrarily large. Because two different expressions satisfy the same differential equation and initially were arbitrarily close and are arbitrarily different in magnitude at any future time, we conclude that the problem is ill posed. Note that, with the proper boundary conditions and initial conditions, equation (24.1) would have a unique solution. But the solution would be unstable because the equation is intrinsically ill posed as an initial value problem. Hence, there would be, for instance, no easy way to numerically approximate the solution.

Notes 1. For a discussion of existence and uniqueness theorems, see page 53. For a discussion of stability theorems, see page 101. 2. A standard example of an ill posed problem is Laplace’s equation with initial data. For example, the equation ∇2 u = 0 with the initial data ∂u 1 1 ∂y (x, 0) = n sin nx has the solution u(x, y) = n2 sin nx sinh ny. As n → ∞, the initial data are becoming arbitrarily small in magnitude whereas the solution (for y > 0) is becoming arbitrarily large. 3. Certain classes of equations have been well studied. We can state • For Laplace’s equation and elliptic equations in general, the Dirichlet problem is well posed. Also, the Neumann problem does not have a unique solution but is otherwise well posed. • For the two-dimensional wave equation and hyperbolic equations in general, both are well posed as an initial value problem. Both are, generally, ill posed as boundary value problems. • For the heat equation and diffusion equations in general, both are well posed when given Dirichlet data and the time variable is increasing; both are ill posed when the time variable is decreasing. See Beck et al. [2] for numerical schemes related to a specific ill posed problem. 4. A backward heat equation (a parabolic equation with decreasing time) is ill posed. It may be made well posed, however, by requiring the solution to satisfy a suitable constraint. Typically, one asks for non-negative solutions or for solutions that satisfy an a priori bound, which is obtained from physical considerations. 5. Payne [9] contains the following non-exhaustive list of methods that have been proposed and used in treating various types of improperly posed Cauchy problems: • Function theoretic methods • Eigenfunction methods • Logarithmic convexity methods c CD-ROM Handbook of Differential Equations Academic Press 1997

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Well Posed Differential Equations • • • • • • • • •

117

Weighted energy methods Lagrange identity methods Quasireversibility methods Restriction of data methods Numerical and programming methods Concavity methods Stochastic and probabilistic methods Method of generalized inverse in reproducing kernel spaces Comparison methods

Payne [9] illustrates several of these methods on a backward heat equation. 6. As Fichera [4] shows, finding the correct boundary conditions for a degenerate problem (one in which the type changes) can be difficult in general. Fichera shows, for example, that the first order equation for u(x, y) a(x, y)ux + b(x, y)uy + cu = f in the rectangle R = {−α ≤ x ≤ α, −β ≤ y ≤ β}, when a and b satisfy a(−α, y) ≥0, b(x, −β) ≥0,

a(α, y) ≤ 0, b(x, β) ≤ 0,

has no boundary conditions! However, the equation, −a(x, y)ux − b(x, y)uy + cu = f, in R, with the same conditions on a and b, requires that u be given on the entire boundary of R. 7. See also Garabedian [5, pages 450–457] and Zauderer [10, pages 103– 113].

References [1] Argyros, I. K. On the cardinality of solutions of multilinear differential equations and applications. Int. J. Math. & Math. Sci. 9, 4 (1986), 757–766. [2] Beck, J. V., Blackwell, B., and St. Clair, Jr., C. R. Inverse Heat Problems. Wiley, New York, 1985. [3] Buzbee, B. L., and Carasso, A. On the numerical computation of parabolic problems for preceding times. Math. of Comp. 27, 122 (April 1973), 237–266. [4] Fichera, G. On a unified theory of boundary value problems for elliptic– parabolic equations of second order. In Boundary Problems in Differential Equations, R. E. Langer, Ed. University of Wisconsin Press, Madison, Wisconsin, 1960, pp. 97–120. [5] Garabedian, P. R. Partial Differential Equations. John Wiley & Sons, New York, 1964. c CD-ROM Handbook of Differential Equations Academic Press 1997

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[6] Lavrent’ev, M. M., Romanov, V. G., and Shishatskii, S. P. IllPosed Problems of Mathematical Physics and Analysis. Amer. Math. Soc., Providence, RI, 1986. [7] Morozov, V. A. Methods for Solving Incorrectly Posed Problems. Springer– Verlag, New York, 1984. [8] Pavlov, N. N. Smoothing of input data in the solution of ill-posed problems. U.S.S.R. Comput. Maths. Math. Phys. 29, 5 (1989), 110–114. [9] Payne, L. E. Improperly Posed Problems in Partial Differential Equations. SIAM, Philadelphia, PA, 1975. [10] Zauderer, E. Partial Differential Equations of Applied Mathematics. John Wiley & Sons, New York, 1983.

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Wronskians and Fundamental Solutions

25.

119

Wronskians and Fundamental Solutions

Applicable to

Linear ordinary differential equations.

Yields A formulation of a linear ordinary differential equation as vector system

Idea An nth order linear ordinary differential equation can be written as a first order ordinary differential equation for a n element vector.

Procedure Let L[·] be the linear nth order ordinary differential operator L[y] =

d(n−1) y dn y + a1 (x) (n−1) + · · · + an (x)y. n dx dx

The vector equation associated with the linear equation L[y] = 0 is given by (see page 146) y0 = A(x)y,  where y = y

y0 

y 00 0 0 0 .. .

    A=    0 −an

...

y (n−1)

T

(25.1)

and A is the matrix

1 0 0 .. .

0 1 0 .. .

0 0 1

0 −an−1

0 −an−2

0 −an−3

··· ··· ..

.

...

0 0 0



    .   1  −a1

(25.2)

If {y1 , y2 , . . . , yn } is any set of n solutions to the equation L[y] = 0, then the matrix   y1 y2 ··· yn  y10 y20 ··· yn0    Φ(x) =  .. ..  .. ..  . . .  . (n−1) (n−1) (n−1) y1 y2 · · · yn is a solution matrix for equation (25.1). It is also called a fundamental solution. This matrix satisfies the differential equation Φ0 = AΦ. The determinant of this matrix, det Φ(x), is called the Wronskian of L[y] = 0 with respect to {y1 , y2 , . . . , yn } and is denoted by W (y1 , y2 , . . . , yn ). Note that the Wronskian is a function of x. c CD-ROM Handbook of Differential Equations Academic Press 1997

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If Φ(x) satisfies Φ0 = AΦ, then |Φ(x)|0 = |Φ| trA(t), where tr A denotes the trace of the matrix A. Hence, Z det Φ(x) = det Φ(x0 ) exp



x

tr A(s) ds . x0

For the matrix in equation (25.2), we have tr A = −a1 so that  Z W (y1 , . . . , yn )(x) = exp −



x

a1 (s) ds W (y1 , . . . , yn )(x0 ).

x0

(25.3)

This is sometimes called Liouville’s formula. From equation (25.3), we conclude that either W (x) vanishes for all values for x, or it is never equal to zero. If the Wronskian never vanishes, then the set {y1 , y2 , . . . , yn } is said to be linearly independent. A set of n linearly independent solutions to L[y] = 0 is called a basis or a fundamental set. Alternately, given a set of n linearly independent continuous functions, {y1 , y2 , . . . , yn }, it is possible to find a unique homogeneous differential equation of order n (with the coefficient of y (n) being one) for which the set forms a fundamental set. This differential equation is given by (−1)n

W (y, y1 , y2 , . . . , yn ) = 0. W (y1 , y2 , . . . , yn )

(25.4)

Example 1 Given the second order linear ordinary differential equation y 00 + y = 0,

(25.5)

the set {sin x, cos x} forms a fundamental set because each element in this set satisfies equation (25.5) and also the Wronskian is given by sin x W (sin x, cos x) = cos x

cos x = −1, − sin x

which does not vanish. Because the Wronskian is constant, we have verified that a1 (x) = 0 in equation (25.5) (the a1 (x) term in this equation corresponds to the first derivative term).

Example 2 If we choose the two functions y1 = sin x and y2 = x, we can determine the linear second order equation that has these solutions as its fundamental c CD-ROM Handbook of Differential Equations Academic Press 1997

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set by constructing equation (25.4). Here, n = 2 so we find y x sin x 0 y 1 cos x 00 y 0 − sin x W (y, x, sin x) , = (−1)2 x sin x W (x, sin x) 1 cos x

=

(x cos x − sin x)y 00 + (x sin x)y 0 − (sin x)y , (x cos x − sin x)

= y 00 +

x sin x sin x y0 − y. (x cos x − sin x) (x cos x − sin x)

Notes 1. Given the linear partial differential equation L[u] =

n X

X ∂u ∂2u + bi + cu ∂xi ∂xj i=1 ∂xi n

aij (x)

i,j=1

for u(x), let Γ = Γ(x, ξ) = Γ(ξ, x) be the geodesic distance between the points p x and ξ. (For a rectangular coordinate system, Γ(x, ξ) = ||x − ξ|| = (x1 − ξ1 )2 + · · · + (xn − ξn )2 .) A fundamental solution, S(x, ξ), satisfies L[S] = 0 and, near x = ξ, has the form S = ΓUm + V log Γ + W , where U , V , and W are analytic functions and m = (n − 2)/2. For example, for Laplace’s equation in n dimensions with n > 2, ∇2 u = 0, a fundamental solution is given by S=

1 rn−2

,

with

r=

p (x1 − ξ1 )2 + · · · + (xn − ξn )2 .

See Garabedian [3, pages 152–153] for details. 2. The canonical form of a self-adjoint third order linear homogeneous differential equation is y 000 + 2Ay 0 + A0 y = 0 (see pages 98 and 163). A fundamental set of solutions for this equation is {u2 , uv, v 2 }, where u(x) and v(x) are any two linearly independent solutions of the second order differential equation u00 + 12 Au = 0. 3. Similar to the second example, it is possible to find a single differential equation whose solutions include the products of the solutions of two given linear homogeneous differential equations; see Spigler [6]. 4. See also Boyce and DiPrima [1, pages 113–126], Coddington and Levinson [2, pages 67–84], Ince [4, pages 116–121], and Simmons [5, pages 76–80].

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References [1] Boyce, W. E., and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, fourth ed. John Wiley & Sons, New York, 1986. [2] Coddington, E. A., and Levinson, N. Theory of Ordinary Differential Equations. McGraw–Hill Book Company, New York, 1955. [3] Garabedian, P. R. Partial Differential Equations. John Wiley & Sons, New York, 1964. [4] Ince, E. L. Ordinary Differential Equations. Dover Publications, Inc., New York, 1964. [5] Simmons, G. F. Differential Equations with Applications and Historical Notes. McGraw–Hill Book Company, New York, 1972. [6] Spigler, R. The linear differential equation whose solutions are the products of solutions of two given differential equations. J. Math. Anal. Appl. 98 (1984), 130–147.

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Zeros of Solutions

26.

123

Zeros of Solutions

Applicable to

Linear ordinary differential equations.

Yields Statements about the zeros of the solutions.

Idea There are several standard theorems about the zeros of solutions of differential equations.

Procedure Consider the following equations: d dx

  dy p(x) + q(x)y = 0 dx

(26.1)

and

and

d2 y + p(x)y = 0 dx2 d2 y + q(x)y = 0 dx2

(26.2.a-b)

  dy d p1 (x) + q1 (x)y = 0 dx dx   dy d p2 (x) + q2 (x)y = 0. dx dx

(26.3.a-b)

1. Consider the self-adjoint equation (26.1) in which p(x) > 0 and p(x) and q(x) are continuous. Sturm’s separation theorem states Theorem Let u and v be linearly independent solutions of (26.1). If α and β are successive zeros of u, then v has one and only one zero in the interval (α, β). This has been extended by Makay [3] to be Theorem Consider the second order equation F (y 00 , y 0 , y, x) = 0,

(26.4)

where F is continuous. If the two conditions are satisfied • If y is a solution of (26.4), then so is cy, for all real c. c CD-ROM Handbook of Differential Equations Academic Press 1997

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• The solution of (26.4) as an initial value problems is unique. then the results of Sturm’s theorem apply to equation (26.4). 2. We have the following result about the interlacing of zeros: Theorem Let u(x) and v(x) be linearly independent solutions of equation (26.2.a) and assume u(x) has at least two zeros in the interval (a, b). Then, if x1 and x2 are two consecutive zeros of u(x), the function v(x) has one, and only one, zero in the interval (x1 , x2 ). Theorem Let p(x) in equation (26.2.a) be continuous in (a, b) with 0 < m ≤ p(x) ≤ M . If the solution u(x) of (26.2.a) has π π two successive zeros x1 and x2 , then √ ≤ x2 − x1 ≤ √ . m M 3. We have the following results about oscillatory solutions: Theorem Consider the self-adjoint equations in (26.3.a-b). If • All the solutions of (26.3.a) are oscillatory as x → ∞. • q2 (x) ≥ q1 (x) are continuous functions, • p2 (x) ≥ p1 (x) > 0 are continuous functions, then all solutions of equation (26.3.b) are oscillatory. Theorem If p(x) ≥ (1 + )/4t2 and  > 0, then all solutions to equation (26.2.a) are oscillatory. Theorem If all the solutions to equation (26.2.a) are oscillatory, and if q(x) ≥ p(x), then all solutions of equation (26.2.b) are oscillatory. And we have the converse: Theorem If q(x) ≥ p(x) and some solutions to equation (26.2.b) are nonoscillatory, then some solutions of equation (26.2.a) must be nonoscillatory. 4. The Sturm comparison theorem is Theorem Consider the self-adjoint equations (26.3.a-b). Let p1 (x) ≥ p2 (x) > 0 and q1 (x) ≥ q2 (x) be continuous functions. Then between any two zeros of a nontrivial solution of equation (26.3.a), there will be at least one zero of every nontrivial solution of (26.3.b). 5. Considering equation (26.1), let p(x) > 0, and let p and q be continuous on [0, ∞]. Then c CD-ROM Handbook of Differential Equations Academic Press 1997

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R ∞ dx R∞ Theorem If 1 p(x) and 1 q(x) dx both diverge, then every solution to equation (26.1) has infinitely many zeros on the interR 1 dx R1 val [1, ∞]. If, in addition, 0 p(x) and 0 q(x) dx both diverge to +∞, then every solution to equation (26.1) has infinitely many zeros on the interval [0, 1]. R x R ∞ dx Theorem If a p(x) converges and if a q(s) ds is bounded by a constant for a ≤ x ≤ ∞, then every non-trivial solution to equation (26.1) has at most a finite number of zeros on the interval [a, ∞]. 6. We also have the following nonoscillation results: Theorem If lim sup x2 p(x) = γ ∗ and lim inf x2 p(x) = γ∗ then the solution of equation (26.2.a) is • Nonoscillatory if γ ∗ < 1/4 • Oscillatory if 1/4 < γ∗ R∞ Theorem For the equations in (26.2): If P (x) = x x p(t) dt, R∞ Q(x) = x x q(t) dt, 0 < Q(x) < P (x), and equation (26.2.a) is nonoscillatory in the wide sense, then equation (26.2.b) is nonoscillatory in the wide sense.
R∞ Theorem Consider (26.2.a) and define lim sup x x p(s) ds = x→∞
R∞ P ∗ and lim inf x x p(s) ds = P∗ then x→∞

• A necessary condition that the solution to equation (26.2.a) be nonoscillatory is that P∗ ≤ 1/4 and P ∗ ≤ 1. • A sufficient condition that the solution to equation (26.2.a) be nonoscillatory is that P ∗ ≤ 1/4.

Notes

1. Makay’s [3] theorem applies to equations such as y 00 (y 0 )2 + y 3 = 0. 2. For the eigenvalue problem L[u] = λn u, let N (λ) count the number of eigenvalues less than λ. In one dimension the asymptotics of N (λ) can be easily determined because the nth eigenfunction has n zeros. For example, for the Schr¨ oedinger equation −∇2 ψn + q(x)ψn = λn ψn   Z 1 N (λn+ ) N (λn− ) 1 1/2 + = [λn − q(x)]+ dx + O 2 2 π n ( y if y ≥ 0 where [y]+ ≡ . The generalization of this formula to k 0 if y < 0 dimensions is (see Newell [4])   Z [1 + o(1)] 1 k/2 . [λn − q(x)]+ dx + O N (λ) = k k/2 n 2 π Γ(k/2 + 1) c CD-ROM Handbook of Differential Equations Academic Press 1997

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Definitions and Concepts

References [1] Banks, S. B. A note on the location of complex zeros of solutions of linear differential equations. Bull. Amer. Math. Soc. 18, 1 (January 1988), 35–38. [2] Erbe, L. H., Kong, Q., and Zhang, B. G. Oscillation Theory for Functional Differential Equations. Marcel Dekker, New York, 1994. [3] Makay, G. A simple proof for Sturm’s separation theorem. Amer. Math. Monthly (March 1992), 218–219. [4] Newell, G. F. Asymptotic distribution of eigenvalues for the multidimensional Schr odinger equation. J. Math. Physics 21, 8 (August 1980), 2193– 2201.

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27.

Transformations

Canonical Forms

Applicable to

The ordinary differential equations:   dy  p e 2q d2 y + f + + r y = 0, + 2 + dx2 x dx x2 x

(27.1)

dy d2 y + (px2 + 2qx + r)y = 0, + 2(e + f x) 2 dx dx

(27.2)

dy d2 y + (c + nx)y = 0, + (b + mx) 2 dx dx

(27.3)

(α + βx)

  d2 y dy , y, x . =F dx2 dx

(27.4)

Idea Each of these equations has certain canonical forms. When approximations and numerical values for these equations are reported in the literature, it is generally for the canonical forms.

Procedure 1 By changing the dependent and independent variables from y = y(x) to v = v(z), via y(x) = νz λ eµz v(z), x = κz, for some choice of the constants {ν, λ, µ, κ}, equation (27.1) will take the form of one of the following four canonical forms:   2 A d2 v + + + B v = 0, dz 2 z2 z   2 d2 v A v = 0, + + dz 2 z2 z   d2 v A + + 1 v = 0, dz 2 z2 A d2 v + 2 v = 0, dz 2 z where A and B are constants. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Procedure 2 By changing the dependent and independent variables from y = y(x) to v = v(z), via 2

y(x) = νeµz+ξz v(z), x = κz + η, for some choice of the constants {ν, µ, ξ, κ, η}, equation (27.2) will take the form of one the following four canonical forms: d2 v dz 2 d2 v dz 2 d2 v dz 2 d2 v dz 2


+ z 2 + J v = 0, − vz = 0, + v = 0, = 0,

where J is a constant.

Procedure 3 By changing the dependent and independent variables, equation (27.3) can be reduced to Weiler’s canonical form (this is also known as a Kummer equation) z

d2 v dv + (b − z) − av = 0. dz 2 dz

(27.5)

The transformation used to produce equation (27.5) from equation (27.3) has several different forms depending on the numerical values of the coefficients in equation (27.3), see Bateman [2] for details.

Procedure 4 A critical point is called a moving critical point (or singularity) if its location depends on the initial conditions for the differential equation (and so the location of the critical point is not fixed solely by the coefficients of the differential equation). For example, the nonlinear differential equation y 00 = (y 0 )2 2y−1 y 2 +1 has the general solution y(x) = tan [log(Ax + B)], where A and B are arbitrary constants. The initial conditions determine A and B and thus determine the location of the singularities of y(x). Given an ordinary differential equation in the form of equation (27.4), if F (y 0 , y, x) is rational in y 0 , algebraic in y, and analytic in x, and if all of the critical points are fixed, then a change of variables of the form y(x) =

az(x) + b , cz(x) + d

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where a, b, c, d, and w are some functions of x, will transform the equation to one of 50 standard forms. Each of these 50 differential equations is for the unknown function z(x). Of these standard forms, six have solutions in terms of the Painlev´e transcendents and all the others have first integrals that are equations of first order or have elementary integrals. The equations that define the six Painlev´e transcendents are d2 y dx2

= 6y 2 + x,

d2 y dx2

d2 y dx2

= 2y 3 + xy + α,  2 dy dy = y1 dx − x1 dx + x1 (αy 2 + β) + γy 3 + yδ ,  2 3 dy 1 = 2y + 3y2 + 4xy 2 + 2(x2 − α)y + βy , dx   2    (y−1)2 dy β 1 1 1 dy αy + = 2y + y−1 − + 2 dx x dx x y +

γy x

d2 y dx2

+ δy(y+1) y−1 ,  1 = 12 y1 + y−1 +

dy dx

d2 y dx2 d2 y dx2

1 y−x



dy dx

2

 α+ + y(y−1)(y−x) 2 2 x (x−1)

− βx y2



1 x

+



1 x−1

+

1 y−x

γ(x−1) (y−1)2

+

δx(x−1) (y−x)2

+

 .

In the above equations, all of the parameters are assumed to be constant.

Notes 1. The first three transformations may be found in Bateman [2, pages 75–79]. 2. The transformations for equation (27.4) may be found in Ince [4, Chapter 14, pages 317–355]. 3. Even though the Painlev´e equations do not have elementary solutions in general, some choices of the parameters will lead to equations solvable in terms of elementary functions. For example, y = −1/x is a solution of the second Painlev´e equation when α = 1, and y = −1/x + 3x2 /(x3 + 4) is a solution of the same equation when α = −2. See Airault [1] for details.

References [1] Airault, H. Rational solutions of Painleve equations. Stud. Appl. Math. 61 (1979), 31–53. [2] Bateman, H. Partial Differential Equations of Mathematical Physics. Dover Publications, Inc., New York, 1944. [3] Berkovitch, L. M. Canonical forms of ordinary linear differential equations. Arch. Math. (Brno) 24, 1 (1988), 25–42. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Canonical Forms

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[4] Ince, E. L. Ordinary Differential Equations. Dover Publications, Inc., New York, 1964. [5] Irvine, D., and Savageau, M. A. Efficient solution of nonlinear ordinary differential equations expressed in S-system canonical form. SIAM J. Numer. Anal. 27, 3 (1990), 704–735. [6] Neuman, F. Transformation and canonical forms of functional–differential equations. Proc. Roy. Soc. Edin. 115A (1990), 349–357.

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28.

Transformations

Canonical Transformations

Applicable to A system of ordinary differential equations that arise from a Hamiltonian. Yields A different system of ordinary differential equations that arise from a different Hamiltonian.

Procedure A Hamiltonian H(p, q), with p = (p1 , . . . , pn ) and q = (q1 , . . . , qn ), defines the system of ordinary differential equations p˙i = − q˙i =

∂H = −Hqi , ∂qi ∂H = Hpi , ∂pi

where a dot denotes differentiation with respect to the independent variable t (see page 61). The {pi , qi } are called the coordinates of the Hamiltonian. The transformation to the new system of coordinates {Pi , Qi } via pi = pi (P, Q), qi = qi (P, Q),

(28.1)

is (commonly) said to be canonical if Hamilton’s equations remain invariant. That is, there exists a new Hamiltonian K(P, Q) such that the equations P˙i = −KQi , Q˙ i = KPi ,

(28.2)

are valid. Canonical transformations can be defined implicitly by a generating function. For instance, for almost arbitrary S(p, Q, t), a canonical transformation is given by Pi = −SQi , qi = −Spi , K(P, Q) = H(p, q) + St ,

(28.3.a-c)

where equations (28.3.a) and (28.3.b) must be solved to obtain explicit expressions for q(P, Q), p(P, Q). Note that, for the St term, the derivative is taken with respect to the explicit dependence of S on t. c CD-ROM Handbook of Differential Equations Academic Press 1997

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Canonical Transformations

133

Other functional forms for the generating function are also possible. For example, a function of the form S(q, P, t) gives rise to the canonical transformation pi = S qi , Qi = S Pi ,

(28.4.a-c)

K(P, Q) = H(p, q) + S t .

Example Given the Hamiltonian H=


1 2 p + a2 (t)q 2 , 2

(28.5)

Hamilton’s equations are {p˙ = −a2 q, q˙ = p}, which can be combined to yield q¨ + a2 q = 0.

(28.6)

Hence, the Hamiltonian in (28.5) defines the second order ordinary differential equation (28.6). Now consider the canonical transformation induced by the generating function S(q, P, t) = q 2 P . From equation (28.4) we find p = 2qP, Q = q2 ,
Q
1 2 p + a2 q 2 = 4P 2 + a2 . K(Q, P ) = 2 2 The equations corresponding to the new Hamiltonian are
1 P˙ = − 4P 2 + a2 , 2 Q˙ = 4P Q.

(28.7.a-b)

Equation (28.7.a) is a nonlinear first order ordinary differential equation for P (t). After P (t) is determined, equation (28.7.b) can be used to determine Q(t) by quadrature. Hence, this change of variable has changed a second order linear ordinary differential equation into two successive first order ordinary differential equations.

Notes 1. Canonical transformations are sometimes called contact transformations. See page 249 for the correct definition of a contact transformation. 2. Technically, and in more generality, a transformation of the 2n variables {xj , pj | j = 1, . . . , n} to the 2n variables {Xj , Pj | j = 1, . . . , n} c CD-ROM Handbook of Differential Equations Academic Press 1997

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Transformations

Pn is a canonical transformation if the differential form j=1 (Pj dXj − pj dxj ) is exact, i.e., there exists a function U = U (x, p) such that n X

(Pj dXj − pj dxj ) = dU.

(28.8)

j=1

3. The section on Hamilton–Jacobi theory (see page 61) utilizes canonical transformations to derive the Hamilton–Jacobi equation. 4. Tolstoy [7] shows that any nonlinear ordinary differential equation may be transformed, in principle, by a variable transformation into a linear differential equation or a system of such equations. This is the reverse of the process that was seen in the example. 5. The set of all canonical transformations forms a group. 6. Fouling transformations are canonical transformations in which the p coordinates in configuration space are preserved (i.e., P = p, Q = Q(p, q)). See Gelman and Saletan [4] for details. 7. A transformation, given by equation (28.1), which allows equation (28.2) to be written, and may or may not satisfy (28.8) is technically called a canonoid transformation. The lack of distinction between canonical and canonoid has occasionally led to ambiguity in the literature. See Currie and Saletan [3] or Negri et al. [6] for details. 8. See also Caratheodory [1, Chapter 6, pages 79–101], Chester [2, pages 197–206], and Goldstein [5, Chapter 8, pages 237–272].

References [1] Caratheodory, C. Calculus of Variations and Partial Differential Equations of the First Order. Holden–Day, Inc., San Francisco, CA, 1965. [2] Chester, C. R. Techniques in Partial Differential Equations. McGraw–Hill Book Company, New York, 1970. [3] Currie, D. G., and Saletan, E. J. Canonical transformations and quadratic Hamiltonians. Nuovo Cimento B 9, 1 (1972), 143–153. [4] Gelman, Y., and Saletan, E. J. q-equivalent particle Hamiltonians. II: The two-dimensional classical oscillator. Nuovo Cimento B 18, 1 (1973), 53–71. [5] Goldstein, H. Classical Mechanics. Addison–Wesley Publishing Co., Reading, MA, 1950. [6] Negri, L. J., Oliveira, L. C., and Teixeira, J. M. Canonoid transformations and constants of the motion. J. Math. Physics 28, 10 (Oct 1987), 2369–2372. [7] Tolstoy, I. Remarks on the linearization of differential equations. J. Inst. Maths. Applics 20 (1977), 53–60.

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29.

Darboux Transformation

29.

135

Darboux Transformation

Applicable to Linear second order ordinary differential equations, a single equation or a system. Yields A reformulation of the problem.

Procedure Given the equation y 00 = (f (x) + κ)y

(29.1)

for y(x), we say that the transformation z(x) = A(x, λ)y + B(x, λ)y 0 is a Darboux transformation if z(x) satisfies a differential equation of the form z 00 = (g(x) + λ)z.

(29.2)

For example, if w(x) is a known solution of equation (29.1), then a Darboux transformation is given by z = y0 − y

w0 . w

(29.3)

In this case, if y satisfies equation (29.1), then z(x) satisfies equation (29.2) with f (x) = g(x) − 2[log w(x)]00 . That is to say, this transformation changes the potential function appearing in equation (29.1) from f (x) by δf = −2[log w(x)]00 , where w(x) is an arbitrary solution of equation (29.1). The usefulness of this technique is that equation (29.2) might be easy to solve for z(x); then y(x) may be found from equation (29.3) by a single integration. For the system of second order ordinary differential equations y00 = D(x)y, where D(x) is the matrix 

(29.4)

d11 (x)  d21 (x)  D(x) =  .  ..

d12 (x) d22 (x) .. .

... ... .. .

 d1n (x) d2n (x)   ..  , . 

dn1 (x)

dn2 (x)

...

dnn (x)

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Transformations

we say that z(x) = A(x)y + B(x)y0 ,

(29.5)

where A and B are matrices, is a Darboux transformation if z satisfies an equation of the form z00 = F (x)z,

(29.6)

where F (x) is a new matrix. Sometimes Darboux transformations of this type can be used to decouple systems of differential equations. See Humi [2] for details.

Example 1 If the solution of the differential equation y 00 = (f (x) + λ)y

(29.7)

is known for each value of λ (call the solutin yλ ), and w(x) = yµ (x) is the solution when λ = µ, then the general solution of the differential equation     d2 1 00 +λ−µ z (29.8) z = w(x) 2 dx w(x) for z(x) is given by (see equation (29.3)) z = yλ0 − yλ

w0 (x) , w(x)

(29.9)

for λ 6= µ. In particular, if we take f (x) √= 0 in equation (29.7), then y0 (x) = Ax + B when λ = 0 and yλ (x) = e± λx for λ 6= 0. If we take µ = 0 and w(x) = x, then equation (29.8) becomes   2 00 z = + λ z, x2 with the solution given by equation (29.9); that is,   √ √ 1 . z(x) = e± λx ± λ − x

Example 2 This example is from Humi [2]. Suppose we wish to decouple a system of symmetric equations in the form of equation (29.4) with   d(x) u1 (x) + λ D(x) = . d(x) u2 (x) + λ c CD-ROM Handbook of Differential Equations Academic Press 1997

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Darboux Transformation

137

If we apply a Darboux transformation, we can hope to obtain the form of equation (29.6) with F (x) given by   0 v (x) + λ F (x) = 1 . (29.10) 0 v2 (x) + λ If we choose B = I in equation (29.5), then to obtain equation (29.6), we require A00 + D0 + AD = F A, 2A0 + D = F. In our case, with D(x) given by equation (29.7) and F (x) given by equation (29.10) we require that the elements of the matrix A(x) satisfy 2a012 = 2a021 = −d, 2a011 + u1 (x) = v1 (x), 2a022 + u2 (x) = v2 (x).

(29.11)

It is a simple matter to integrate these equations to obtain a12 (x) =a21 (x) = c(x),   1 1 d(x) + α + I , a11 (x) = 2c 2   1 1 a22 (x) = d(x) − α + I , 2c 2 where α is an arbitrary constant and Z 1 x c(x) = − d(t) dt, 2 Z x c(t)[u2 (t) − u1 (t)] dt. I(x) = This solution is valid if the consistency constraint  2  0 1 d 1 d 2 + + 2 (α + I)2 u1 + u2 = 2c − 2c 2 2c 2c

(29.12)

is satisfied. This constraint was derived in the solution of equation (29.11). Stated another way, we can choose d and u1 − u2 as arbitrary functions and then use equation (29.12) to compute the corresponding u1 + u2 for which the resulting system of equations can be decoupled by the use of a Darboux transformation.

Note 1. See Ince [3, page 182] and Lamb [5, pages 38–41].

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References [1] Darboux, G. C. R. Acad. Sci Paris (1882), 1456. [2] Humi, M. Factorization of systems of differential equations. J. Math. Physics 27 (Jan 1986), 76–81. [3] Ince, E. L. Ordinary Differential Equations. Dover Publications, Inc., New York, 1964. [4] Konopelchenko, B. G. On exact solutions of nonlinear integrable equations via integral linearising transforms and generalised Backlund– Darboux transformations. J. Phys. A: Math. Gen. 23 (1990), 3761–3768. [5] Lamb, G. L. Elements of Soliton Theory. John Wiley & Sons, New York, 1980. [6] Levi, D. Toward a unification of the various techniques used to integrate nonlinear partial differential equations: Backlund and Darboux transformations vs. dressing method. Rep. Math. Phys. 23, 1 (1986), 41–56. [7] Poplavskii, I. V. Generalized Darboux–Crum–Krein transformations. Theo. Math. Physics 69, 3 (1986), 1278–1282. [8] Sall, M. A. Darboux transformations for non-Abelian and nonlocal equations of the Toda chain type. Theo. Math. Physics 53 (1982), 1092– 1099. [9] Stanek, S., and Vosmansky, J. Transformations of linear second order ordinary differential equations. Archivum Mathematicum (BRNO) 22, 1 (1986), 55–60. [10] Zheng, W. M. The Darboux transformation and solvable double-well potential models for Schrodinger equations. J. Math. Physics 25, 1 (Jan 1984), 88–90.

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30.

An Involutory Transformation

30.

139

An Involutory Transformation

Applicable to

Nonlinear partial differential equations of a certain

form.

Yields A reformation of the partial differential equation.

Idea Inverting the dependent and independent variables might lead to a more tractable equation.

Procedure Suppose we have a partial differential equation of the form   ∂ ∂ ; := Φ(u, ux , uxx , . . . ; ut , utt , . . . ) = 0, Φ u; ∂x ∂t

(30.1)

for u = u(x, t). We introduce the inverse transformation  0   u = x, T = x0 = u,   0 t = t. Because applying T twice is equivalent to not applying T , the transformation is involutory (i.e., T 2 = I = the identity). Noting that 1 ∂ ∂ = ∂x ∂u0 /∂x0 ∂x0 ∂ ∂ ∂u0 /∂t0 ∂ = 0− ∂ 0 := , ∂t ∂t ∂u0 /∂x0 ∂x0

D0 :=

then, under T , equation (30.1) becomes Φ (x; D0 ; ∂ 0 ) = 0.

(30.2)

This transformation may be used to change classes of nonlinear equations with Dirichlet boundary conditions to linear form. For example, the class ∂ ∂u0 − γ(u0 ) 0 ∂t0 ∂x

N X

! 0i 0

αi (u0 , t0 )D x

= 0,

i=1

u0 = Ψ1 (t0 ) u0 = Ψ2 (t0 )

on x0 = Φ1 (t0 ), on x0 = Φ2 (t0 ),

u0 = Θ(x0 )

at t0 = 0,

(30.3)

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Transformations

transforms, under T , to N X

∂iu αi (x, t) i ∂x i=1

∂ ∂u + γ(u) ∂t ∂x u = Φ1 (t)

on x = Ψ1 (t),

u = Φ2 (t) u = Θ−1 (x)

on x = Ψ2 (t), at t = 0.

! = 0, (30.4)

Example Given the equation and initial/boundary conditions ∂u0 = ∂t0

κ


∂u0 2 ∂x0

u0 = 0 u0 = L u0 = Θ(x0 )

∂ 2 u0 , ∂x0 2 on x0 = Φ1 (t0 ), on x0 = Φ2 (t0 ),

(30.5)

at t0 = 0,

the transformed equation and initial/boundary conditions become ∂2u ∂u = κ 2, ∂t ∂x u = Φ1 (t) on x = 0, on x = L, u = Φ2 (t) u = Θ−1 (x)

(30.6)

at t = 0.

Then equation (30.6) can be easily solved (by use of, say, Fourier transforms) to yield "   ∞  nπx  Z L  nπσ  κn2 π 2 t 2 X −1 dσ sin exp − Θ (σ) sin u(x, t) = L n=1 L2 L L 0  2 2   Z nκπ t κn π τ n + [Φ1 (τ ) − (−1) Φ2 (τ )] dτ . exp L 0 L2 This last relation, can be implicitly solved for x = x(u, t); which (under T ) is the solution to equation (30.5) (i.e., u0 = u0 (x0 , t0 )).

Note 1. The Hodograph transformation is a different way in which the dependent and independent variables are interchanged (see page 456).

Reference [1] Rogers, C. Inverse transformations and the reduction of nonlinear Dirichlet problems. J. Phys. A: Math. Gen. 17 (1984), L681–L685. c CD-ROM Handbook of Differential Equations Academic Press 1997

31.

Liouville Transformation – 1

31.

141

Liouville Transformation – 1

Applicable to

The general Sturm–Liouville problem

−[p(x)y 0 ]0 + r(x)y = λρ(x)y, y 0 (a) + αy(a) = 0,

for a ≤ x ≤ b, (31.1)

y 0 (b) + βy(b) = 0.

Procedure The Liouville transformation (version 1) is to change the independent variable from x ∈ [a, b] to t ∈ [0, π] by 1/2 Z  1 x ρ(x) dx, (31.2) t= J a p(x) where J is defined by 1 J= π

Z a

b



ρ(x) p(x)

1/2 dx,

(31.3)

and to change the dependent variable from y(x) to u(t) by u(t) = f (x)y(x) = [ρ(x)p(x)]1/4 y(x),

(31.4)

where we have defined f (x) = [ρ(x)p(x)]1/4 . With this change of variable, equation (31.1) becomes d2 u + [k 2 − q(t)]u = 0, dt2 u0 (0) + hu(0) = 0, u0 (π) + Hu(π) = 0,

for 0 ≤ t ≤ π, (31.5)

which is in Liouville normal form. The definitions of {k, q(t), h, H} are as follows k 2 = J 2 λ, r(x) , ρ(x) ftt + J 2 m(t), q(t) = f 1 h = 2 [αJp(a) − f (a)ft (a)], f (a) 1 H = 2 [βJp(b) − f (b)ft (b)]. f (b)

m(t) =

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Note that q(t) may also be written as r d2 + (pρ)−1/4 2 [(pρ)1/4 ], p dt "   2   0  2 #  0 0 0 0 p 1 ρ0 r 3 p0 1 p0 p ρ ρ − = + + + + p 4ρ p ρ 4 p 2 p ρ 4 ρ

q(t) =

Example If we have the equation and boundary conditions 1 y = λxy, x y 0 (π) = 0,

−(xy 0 )0 +

for π ≤ x ≤ 2π,

y 0 (2π) = 0, then we identify p(x) = x, a = π,

1 , ρ(x) = x, x b = 2π, α = 0,

r(x) =

β = 0.

√ √ A simple calculation results in J = 1, t = x − π, f (x) = x = t + 1, 1 3 1 2 m(t) = x12 = (t+1) = λ, h = − 21 , and H = − 2(π+1) . 2 , q(t) = 4(t+1)2 , k Hence, we obtain   3 00 u = 0, for 0 ≤ t ≤ π, u + λ− 4(t + 1)2 1 (31.6) u0 (0) − u(0) = 0, 2 1 u(π) = 0. u0 (π) − 2(π + 1) Equation (31.6) is in Liouville normal form.

Notes 1. The standard assumptions used with equation (31.1) are that on the interval [a, b]: p and q are real-valued, p > 0, q does not vanish, and p and q have continuous second derivatives. Boundedness conditions are also required for the new functions. 2. When ρ = 0, the transformation Z xs |q(z)| dz, t= p(z) (31.7) x0 u(t) = [p(x)|q(x)|]

1/4

y(x),

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31.

Liouville Transformation – 1

143

when applied to equation (31.1), results in d2 u + [±1 + R(t)] u(t) = 0, dt2

(31.8)

where R(t) = p1/4 |q|−3/4

dp(x) d −1/4 [p(x)|q(x)|] , x=x(t) dx dx

and the plus (minus) sign is taken in equation (31.8) if q(x) > 0 (q(x) < 0). This is also called the Liouville transformation (see Eastham [4]). 3. The two different transformations, the one in equations (31.2) and (31.4), and the one in equation (31.7), are each sometimes called the Liouville–Green transformation. 4. See also Birkhoff and Rota [1, pages 265–267], Boyce [2, pages 20–21], Hille [5, page 340], Lakin and Sanchez [7, pages 36–41], and Valiron [8, page 511].

References [1] Birkhoff, G., and Rota, G.-C. Ordinary Differential Equations. John Wiley & Sons, New York, 1978. [2] Boyce, W. E. Random eigenvalue problems. In Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, Ed. Academic Press, New York, 1968, pp. 1–73. [3] Cassell, J. S. An extension of the Liouville–Green asymptotic formula for oscillatory second-order differential equations. Proc. Roy. Soc .Edin. 100A (1985), 181–190. [4] Eastham, M. S. P. Asymptotic formulae of Liouville–Green type for higherorder differential equations. J. London Math. Soc. 2, 28 (1983), 507–518. [5] Hille, E. Lectures on Ordinary Differential Equations. Addison–Wesley Publishing Co., Reading, MA, 1969. [6] Howard, H. C., and Maric, V. An extension of the Liouville–Green approximation. J. Math. Anal. Appl. 143, 2 (1989), 548–559. [7] Lakin, W. D., and Sanchez, D. A. Topics in Ordinary Differential Equations. Dover Publications, Inc., New York, 1970. [8] Valiron, G. The Geometric Theory of Ordinary Differential Equations and Algebraic Functions. Math Sci Press, Brookline, MA, 1950.

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32.

Transformations

Liouville Transformation – 2

Applicable to

The second order linear ordinary differential equa-

tion d2 y + λm4 (t)y = 0 dt2

(32.1)

on the finite interval 0 ≤ t ≤ T , where λ is a constant and m(t) > 0.

Procedure The Liouville transformation (version 2) is to change the dependent and independent variables in equation (32.1) by 1 x= J

Z

t

m2 (z) dz, 0 Z T

1 m2 (z) dz, π 0 w(x) = m(t)y(t). J=

This transformation changes equation (32.1) into  d2 w  2 + λJ + Q(x) w = 0, 2 dx

(32.2)

for 0 ≤ x ≤ π, where Q(x) is defined by Q(x) =

J 2 d2 1 d2 m(t) =− 2 m(t) dx m(t)3 dt2



1 m(t)

 .

(32.3)

The inverse transformation, which takes equation (32.2) into equation (32.1), is given by Z x dζ t=J , ∗ (ζ)]2 [m 0 −1 Z π dζ J =T , ∗ 2 0 [m (ζ)] where m∗ (x) = m(t) is any positive solution of the differential equation d2 m∗ = Q(x)m∗ (x). dx2 c CD-ROM Handbook of Differential Equations Academic Press 1997

32.

Liouville Transformation – 2

145

Example Suppose we have (essentially) Airy’s equation d2 y + λty = 0. dt2

(32.4)

Comparing equation (32.4) to equation (32.1) shows that m(t) = t1/4 . Using this value for m(t) produces 2 3/2 T , 3π  3/2 t , x=π T

J=

w(x) = t1/4 y(t). Under this change of variables, equation (32.4) becomes   5 1 d2 w 4λ 3 w = 0. + T − dx2 9π 2 36 x2

(32.5)

For large values of x, an approximation to equation (32.5) might be obtained by discarding the second term in the parentheses.

Notes 1. The function Q(x) defined in equation (32.3) will be a constant if and only if m(t) = (αt2 +βt+δ)−1/2 . In this case, Q(x) = −J 2 (αδ −4β 2 ). 2. This transformation is useful when followed by some sort of asymptotic analysis. When the magnitude of λ is large compared to Q(x), then the first order approximation to equation (32.2) will be to discard the Q(x) term. 3. See Magnus and Winkler [1, page 51].

Reference [1] Magnus, W., and Winkler, S. Hill’s Equation. Dover Publications, Inc., New York, 1966.

c CD-ROM Handbook of Differential Equations Academic Press 1997

146

I.B

33.

Transformations

Reduction of Linear ODEs to a First Order System

Applicable to

Linear ordinary differential equations.

Yields A first order vector system.

Idea By introducing variables to represent the derivatives in an nth order linear ordinary differential equation, a first order system of differential equations may be obtained.

Procedure Given the linear ordinary differential equation d(n−1) y dy dn y + a0 (x)y + b(x) = a (x) + · · · + a1 (x) n−1 n (n−1) dx dx dx

(33.1)

for y(x), introduce the variables {z1 , z2 , . . . , zn } defined by z1 =

dy , dx

z2 =

d2 y , dx2

... ,

zn =

dn y . dxn

Using these new variables, equation (33.1) may be written as d y = A(x)y + b(x), dx

(33.2)

where  y= y  b= 0

y (1) 0

...

...

and A is the matrix  0  0   0   ..  .   0 a0 (x)

T  y (n−1) = y T 0 b(x) ,

1 0 0 .. .

0 1 0 .. .

0 0 1

0 a1 (x)

0 a2 (x)

0 a3 (x)

z2

z1

··· ··· ..

...

zn−1

0 0 0

.

...

1 an−1 (x)

T

,

     .   

c CD-ROM Handbook of Differential Equations Academic Press 1997

33.

Reduction of Linear ODEs to a First Order System

147

If the initial conditions for equation (33.1) were in the form y(x0 ) = c0 , y 0 (x0 ) = c1 , y 00 (x0 ) = c2 , . . . , y (n−1) (x0 ) = cn−1 , T  then the initial condition for equation (33.2) is y(x0 ) = c0 c1 . . . cn−1 . To solve an equation in the form of equation (33.2), see the section on vector ordinary differential equations (page 421).

Example Given the linear ordinary differential equation with initial conditions dy d2 y + (log x)y = sin x, + x2 dx2 dx y(0) = 3, y 0 (0) = 4, it may easily be changed into the equivalent first order system        d